MAP/P H/
1QUEUE WITH DISCARDING CUSTOMERS
HAVING IMPERFECT SERVICE
Sindhu S
Department of Mathematics, Model Engineering College. Ernakulam (India).
E-mail:sindhuramachandran99@gmail.com
https://orcid.org/0000-0001-6031-0973
Achyutha Krishnamoorthy
Centre for Research in Mathematics, CMS College. Kottayam (India).
Department of Mathematics, Central University of Kerala. Kasaragod (India).
E-mail:achyuthacusat@gmail.com
https://orcid.org/0000-0002-5911-1325
Reception: 14/09/2022 Acceptance: 29/09/2022 Publication: 29/12/2022
Citación sugerida:
Sindhu S. and Achyutha Krishnamoorthy (2022).
MAP/P H/
1queue with discarding customers having imperfect service.
3C Empresa. Investigación y pensamiento crítico, 11 (2), 116-137.
. https://doi.org/10.17993/3cemp.2022.110250.116-137
https://doi.org/10.17993/3cemp.2022.110250.116-137
ABSTRACT
In this paper, we consider two queueing models. Model I is on a single-server queueing system in which
the arrival process follows MAP with representation
D
=(
D0,D
1
)of order m and service time follows
phase-type distribution (
β
β
β,S
)of order n. When a customer enters into service, a generalized Erlang
clock is started simultaneously. The clock has
k
stages. The
pth
stage parameter is
θp
for 1
≤p≤k
. If
a customer completes the service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of
the clock, it is a perfect one. On the other hand, if the service gets completed either before the
kth
1
stage
realization or after the
kth
2
stage realization, it is discarded because of imperfection. We analyse this model
using the matrix-geometric method. We obtain the expected service time and expected waiting time of a
tagged customer. Additional performance measures are also computed. We construct a revenue function
and numerically analyse it. In Model II, a single server queueing system in which all assumptions are
the same as in Model I except the assumption on service time, is considered. Up to stage
k1
service time
follows phase-type distribution (
α′
α′
α′,T′
)of order
n1
and beyond stage
k1
, the service time follows phase
type distribution (
β′
β′
β′,S′
)of order
n2
. We compare the values of the revenue function of the two models
KEYWORDS
Markovian Arrival Process, Phase-type distribution, Erlang Clock, Imperfect Service.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
116
MAP/P H/
1QUEUE WITH DISCARDING CUSTOMERS
HAVING IMPERFECT SERVICE
Sindhu S
Department of Mathematics, Model Engineering College. Ernakulam (India).
E-mail:sindhuramachandran99@gmail.com
https://orcid.org/0000-0001-6031-0973
Achyutha Krishnamoorthy
Centre for Research in Mathematics, CMS College. Kottayam (India).
Department of Mathematics, Central University of Kerala. Kasaragod (India).
E-mail:achyuthacusat@gmail.com
https://orcid.org/0000-0002-5911-1325
Reception: 14/09/2022 Acceptance: 29/09/2022 Publication: 29/12/2022
Citación sugerida:
Sindhu S. and Achyutha Krishnamoorthy (2022).
MAP/P H/
1queue with discarding customers having imperfect service.
3C Empresa. Investigación y pensamiento crítico, 11 (2), 116-137.
. https://doi.org/10.17993/3cemp.2022.110250.116-137
https://doi.org/10.17993/3cemp.2022.110250.116-137
ABSTRACT
In this paper, we consider two queueing models. Model I is on a single-server queueing system in which
the arrival process follows MAP with representation
D
=(
D0,D
1
)of order m and service time follows
phase-type distribution (
β
β
β,S
)of order n. When a customer enters into service, a generalized Erlang
clock is started simultaneously. The clock has
k
stages. The
pth
stage parameter is
θp
for 1
≤p≤k
. If
a customer completes the service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of
the clock, it is a perfect one. On the other hand, if the service gets completed either before the
kth
1
stage
realization or after the
kth
2
stage realization, it is discarded because of imperfection. We analyse this model
using the matrix-geometric method. We obtain the expected service time and expected waiting time of a
tagged customer. Additional performance measures are also computed. We construct a revenue function
and numerically analyse it. In Model II, a single server queueing system in which all assumptions are
the same as in Model I except the assumption on service time, is considered. Up to stage
k1
service time
follows phase-type distribution (
α′
α′
α′,T′
)of order
n1
and beyond stage
k1
, the service time follows phase
type distribution (
β′
β′
β′,S′
)of order
n2
. We compare the values of the revenue function of the two models
KEYWORDS
Markovian Arrival Process, Phase-type distribution, Erlang Clock, Imperfect Service.
https://doi.org/10.17993/3cemp.2022.110250.116-137
117
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
1 INTRODUCTION
Queueing models play an important role in our everyday life. Important application areas of queueing
models are production systems, transportation and stocking systems, communication systems, infor-
mation processing systems, etc. In a manufacturing system, a product goes through several stages to
getting processed; the processing time of a product is very important.
Phase type distribution was introduced by Neuts (1975) as a generalization of the exponential
distribution . Phase type distribution is defined as the distribution of time to absorption of a Markov
chain with finite transient states and one absorbing state. Let
¯
X
=
{X
(
t
):
t≥
0
}
denote a continuous
time Markov chain with state space
S
=
{
1
,
2
,
3
,...,m,m
+1
}
where the first m states are transient
and the last state is absorbing and with infinitesimal generator matrix
˜
Q
=
TT
0
00
,where
T
is a square matrix of order
m
and
T0
is a column vector and
T0
=
−Te
e
e
.
The initial probability distribution of
¯
X
is
¯
α
α
α
=(
α
α
α, αm+1
)where
α
α
α
is a row vector of dimension
m
and
αm+1
=1
−αe
αe
αe
. Let
Z
=
inf{t≥
0:
X
(
t
)=
m
+1
}
be a random variable of time until absorption in
state
m
+1. The distribution of Z is called a continuous phase-type distribution (PH distribution) with
parameter (
α
α
α, T
). The distribution function of a continuous phase type distribution
PH
(
α
α
α, T
)is given
by
F
(
t
)=1
−α
α
αeTt
e
e
e
for
t≥
0and probability density function is
f
(
t
)=
α
α
αeTtT0
for
t≥
0. The Laplace
Stieltjes transform of
PH
(
α
α
α, T
)is given by
ϕ
(
s
)=
αm+1
+
α
α
α
(
sI −T
)
−1T0
for all
s∈C
with
Re
(
s
)
≥
0.
The Markovian Arrival Process(MAP) was introduced by David M. Lucantoni (1990) as a simpler
version of an earlier model proposed by Neuts (1979). It is a generalization of the Markov process
where arrivals are governed by an underlying m-states Markov chain. A continuous time Markov chain
{
(
N
(
t
)
,J
(
t
)) :
t≥
0
}
with state space
{
(
i, j
):
i
=0
,
1
,
2
,...
;1
≤j≤m}
and infinitesimal generator
matrix
¯
Q=
D0D1
D0D1
D0D1
......
is called a MAP with matrix representation (D0,D
1).
D0
and
D1
are square matrices of order m.
N
(
t
)counts the number of arrivals during (0
,t
)and
J
(
t
)represents the phase of the arrival process.
D0
has negative diagonal elements and non-negative
off-diagonal elements, and its elements correspond to state transition without an arrival.
D1
is a
non-negative matrix whose elements represent state transition with one arrival. Let the matrix
D
be
defined as
D
=
D0
+
D1
. Then
D
is an irreducible infinitesimal generator of the underlying Markov
chain {J(t)}. Let π
π
πbe the invariant probability vector of D, then
π
π
πD
=0
,π
π
πe
e
e
=1. The average rate of events in a
MAP
, which is called the fundamental rate of the
MAP , is given by λ=π
π
πD1e
e
e.
The arrival of a negative customer to a queueing system causes the removal of one ordinary customer
(called a positive customer) who is present in the queue. But the Negative arrivals have no effect if the
system is empty. We can therefore represent a Negative customer as a type of work canceling signal.
Queues with negative arrivals were first introduced by Gelenbe (1991a). So queues with negative arrivals
are called G-queues. Those who are interested in a comprehensive analysis of G-queues may refer to
Gelenbe et al. (1991b), Artalejo (2000), and Bocharov and Vishnevskii (2003).
Valentina Klimenok and Alexander Dudin (2012) consider a multi-server queueing system with finite
and infinite buffers. The input flow is described by Batch Markovian Arrival Process(BMAP) and
the service time has the PH distribution. Besides positive customers, the negative customers arrive
according to the Markovian Arrival Process. A negative customer can remove an ordinary customer in
service if the state service process does not belong to protected phases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
S R Chakravarthy (2009) has considered a single server queueing system in which arrivals occur
according to a Markovian arrival process. All the customers in the system are lost when the system
undergoes disastrous failures. In G-queues a regular customer is pushed out of the system by a negative
customer. But here we consider a queueing system in which a customer is discarded if his service
completion is not within a stipulated time interval.
The queueing models considered so far in the literature did not look at the possibility of service
completion of customers before a threshold or beyond a second threshold. Several real-life situations
warrant the completion of services between the lower and upper thresholds. This is necessitated by the
fact that the raw material used for the production of a specified item may not get completely processed
if completed before time. Similarly, it could get over-processed if the processing completion time gets
beyond a threshold. The subject matter of this paper addresses this important aspect in production
and manufacturing.
This Queueing model can be applied in various fields in our day-to-day life. For example, in a food
manufacturing unit, the correct baking time of a product is a crucial factor. If the baking time exceeds
a threshold, the product gets burnt. On the other hand, if the baking time is not sufficient, the product
will only be half cooked and will not be acceptable.
Another example is the manufacturing of Nylon wires and films. In the manufacture of nylon,
caprolactam (a chemical used as raw material), is melted and the molten caprolactam is catalytically
polymerized at previously optimized conditions of temperature, pressure, the concentration of the
catalyst, etc. Further, the output of the above process is subjected to another process like extrusion or
calendering. Extrusion is used to produce nylon wires, whereas calendering is used to produce nylon
films. The condition of this is also an optimized one, in which any variation will cause defective wires
and films which will not be suitable for end-use. The condition is optimized based on laboratory and
pilot plant situations.
In this paper, we first consider a single-server queueing system in which the arrival process follows
MAP and service time follows the continuous phase-type distribution. When a customer enters into
service, a generalized Erlang clock is started simultaneously. The clock has
k
stages. The
pth
stage
parameter is
θp
for 1
≤p≤k
. If a customer completes the service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of the clock, the final product is perfect. If it gets completed either before
the kth
1stage realization or after the kth
2stage realization, it has to be discarded.
Salient features of this paper are
•it deviates from the classical assumption of merely specifying a service time distribution.
•the lower and upper thresholds for service are the most important additions.
•When a customer enters into service, a generalized Erlang clock is started simultaneously.
•
If a customer completes service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
)
of the Erlang clock, it is perfect.
•
If a customer completes the service either before the
kth
1
stage realization or after the
kth
2
stage
realization, it is discarded.
•
To maximise revenue, in Model II we consider the service time as phase-type distributed with
representation (
γ′
γ′
γ′,L
)of order
n
=
n1
+
n2
, which is the convolution of the two phase type
distributions (α′
(α′
(α′,T′)of order n1and (β′
(β′
(β′,S′)of order n2.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
118
1 INTRODUCTION
Queueing models play an important role in our everyday life. Important application areas of queueing
models are production systems, transportation and stocking systems, communication systems, infor-
mation processing systems, etc. In a manufacturing system, a product goes through several stages to
getting processed; the processing time of a product is very important.
Phase type distribution was introduced by Neuts (1975) as a generalization of the exponential
distribution . Phase type distribution is defined as the distribution of time to absorption of a Markov
chain with finite transient states and one absorbing state. Let
¯
X
=
{X
(
t
):
t≥
0
}
denote a continuous
time Markov chain with state space
S
=
{
1
,
2
,
3
,...,m,m
+1
}
where the first m states are transient
and the last state is absorbing and with infinitesimal generator matrix
˜
Q
=
TT
0
00
,where
T
is a square matrix of order
m
and
T0
is a column vector and
T0
=
−Te
e
e
.
The initial probability distribution of
¯
X
is
¯
α
α
α
=(
α
α
α, αm+1
)where
α
α
α
is a row vector of dimension
m
and
αm+1
=1
−αe
αe
αe
. Let
Z
=
inf{t≥
0:
X
(
t
)=
m
+1
}
be a random variable of time until absorption in
state
m
+1. The distribution of Z is called a continuous phase-type distribution (PH distribution) with
parameter (
α
α
α, T
). The distribution function of a continuous phase type distribution
PH
(
α
α
α, T
)is given
by
F
(
t
)=1
−α
α
αeTt
e
e
e
for
t≥
0and probability density function is
f
(
t
)=
α
α
αeTtT0
for
t≥
0. The Laplace
Stieltjes transform of
PH
(
α
α
α, T
)is given by
ϕ
(
s
)=
αm+1
+
α
α
α
(
sI −T
)
−1T0
for all
s∈C
with
Re
(
s
)
≥
0.
The Markovian Arrival Process(MAP) was introduced by David M. Lucantoni (1990) as a simpler
version of an earlier model proposed by Neuts (1979). It is a generalization of the Markov process
where arrivals are governed by an underlying m-states Markov chain. A continuous time Markov chain
{
(
N
(
t
)
,J
(
t
)) :
t≥
0
}
with state space
{
(
i, j
):
i
=0
,
1
,
2
,...
;1
≤j≤m}
and infinitesimal generator
matrix
¯
Q=
D0D1
D0D1
D0D1
......
is called a MAP with matrix representation (D0,D
1).
D0
and
D1
are square matrices of order m.
N
(
t
)counts the number of arrivals during (0
,t
)and
J
(
t
)represents the phase of the arrival process.
D0
has negative diagonal elements and non-negative
off-diagonal elements, and its elements correspond to state transition without an arrival.
D1
is a
non-negative matrix whose elements represent state transition with one arrival. Let the matrix
D
be
defined as
D
=
D0
+
D1
. Then
D
is an irreducible infinitesimal generator of the underlying Markov
chain {J(t)}. Let π
π
πbe the invariant probability vector of D, then
π
π
πD
=0
,π
π
πe
e
e
=1. The average rate of events in a
MAP
, which is called the fundamental rate of the
MAP , is given by λ=π
π
πD1e
e
e.
The arrival of a negative customer to a queueing system causes the removal of one ordinary customer
(called a positive customer) who is present in the queue. But the Negative arrivals have no effect if the
system is empty. We can therefore represent a Negative customer as a type of work canceling signal.
Queues with negative arrivals were first introduced by Gelenbe (1991a). So queues with negative arrivals
are called G-queues. Those who are interested in a comprehensive analysis of G-queues may refer to
Gelenbe et al. (1991b), Artalejo (2000), and Bocharov and Vishnevskii (2003).
Valentina Klimenok and Alexander Dudin (2012) consider a multi-server queueing system with finite
and infinite buffers. The input flow is described by Batch Markovian Arrival Process(BMAP) and
the service time has the PH distribution. Besides positive customers, the negative customers arrive
according to the Markovian Arrival Process. A negative customer can remove an ordinary customer in
service if the state service process does not belong to protected phases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
S R Chakravarthy (2009) has considered a single server queueing system in which arrivals occur
according to a Markovian arrival process. All the customers in the system are lost when the system
undergoes disastrous failures. In G-queues a regular customer is pushed out of the system by a negative
customer. But here we consider a queueing system in which a customer is discarded if his service
completion is not within a stipulated time interval.
The queueing models considered so far in the literature did not look at the possibility of service
completion of customers before a threshold or beyond a second threshold. Several real-life situations
warrant the completion of services between the lower and upper thresholds. This is necessitated by the
fact that the raw material used for the production of a specified item may not get completely processed
if completed before time. Similarly, it could get over-processed if the processing completion time gets
beyond a threshold. The subject matter of this paper addresses this important aspect in production
and manufacturing.
This Queueing model can be applied in various fields in our day-to-day life. For example, in a food
manufacturing unit, the correct baking time of a product is a crucial factor. If the baking time exceeds
a threshold, the product gets burnt. On the other hand, if the baking time is not sufficient, the product
will only be half cooked and will not be acceptable.
Another example is the manufacturing of Nylon wires and films. In the manufacture of nylon,
caprolactam (a chemical used as raw material), is melted and the molten caprolactam is catalytically
polymerized at previously optimized conditions of temperature, pressure, the concentration of the
catalyst, etc. Further, the output of the above process is subjected to another process like extrusion or
calendering. Extrusion is used to produce nylon wires, whereas calendering is used to produce nylon
films. The condition of this is also an optimized one, in which any variation will cause defective wires
and films which will not be suitable for end-use. The condition is optimized based on laboratory and
pilot plant situations.
In this paper, we first consider a single-server queueing system in which the arrival process follows
MAP and service time follows the continuous phase-type distribution. When a customer enters into
service, a generalized Erlang clock is started simultaneously. The clock has
k
stages. The
pth
stage
parameter is
θp
for 1
≤p≤k
. If a customer completes the service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of the clock, the final product is perfect. If it gets completed either before
the kth
1stage realization or after the kth
2stage realization, it has to be discarded.
Salient features of this paper are
•it deviates from the classical assumption of merely specifying a service time distribution.
•the lower and upper thresholds for service are the most important additions.
•When a customer enters into service, a generalized Erlang clock is started simultaneously.
•
If a customer completes service in between the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
)
of the Erlang clock, it is perfect.
•
If a customer completes the service either before the
kth
1
stage realization or after the
kth
2
stage
realization, it is discarded.
•
To maximise revenue, in Model II we consider the service time as phase-type distributed with
representation (
γ′
γ′
γ′,L
)of order
n
=
n1
+
n2
, which is the convolution of the two phase type
distributions (α′
(α′
(α′,T′)of order n1and (β′
(β′
(β′,S′)of order n2.
https://doi.org/10.17993/3cemp.2022.110250.116-137
119
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Notations and abbreviations used
Notations and abbreviations used
Notations and abbreviations used
•LIQBD: Level independent Quasi-Birth and Death.
•MAP : Markovian Arrival Process.
•CTMC: Continuous time Markov chain.
•IP: Identity matrix of order P.
•ea
ea
ea: Column vector of 1′s of order a.
•e
e
e: Column vector of 1′s of appropriate order.
•x′
x′
x′: Transpose of a vector x
x
x.
The remaining part of this paper is organized as follows. In section 2 the model under study is
mathematically formulated. In section 3 we perform the steady-state analysis of the queueing model.
Service time analysis and waiting time analysis of a customer are discussed in sections 4 and 5
respectively. Some additional performance measures are provided in section 6. A revenue function is
discussed in section 7. Model description and mathematical formulation of model 2 are given in section
8. In section 9 we perform the steady state analysis of model 2. Numerical results are discussed in
section 10.
2 Mathematical formulation of Model I
We consider a single-server queueing system in which the arrival process follows MAP with representation
D
=(
D0,D
1
)of order m and service time follows continuous phase-type distribution (
β
β
β,S
)of order n.
When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has
k
stages. The
pth
stage parameter is
θp
for 1
≤p≤k
. If a customer completes the service in between
the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of the clock, it is perfect. If a customer completes
the service either before the
kth
1
stage realization or after the
kth
2
stage realization, it is discarded. The
expected service rate is
µ
=[
β
β
β
(
−S
)
−1e
]
−1
. Let
D
=
D0
+
D1
be the infinitesimal generator matrix of
the arrival process and
δ
δ
δ
be its stationary probability vector, then
δ
δ
δD
=0
,δ
δ
δe
=1
.
The constant
λ
=
δ
δ
δD1ereferred to as the fundamental rate, gives the expected number of arrivals per unit of time.
2.1 The QBD process
The model described in section 1 can be studied as a LIQBD process. First, we define the following
notations:
N(t): number of customers in the system at time t,
J(t)=j, if the Erlang clock is in the jth stage at time t,j=1,2, ..., k2,
Is(t): the phase of service process at time t,
Ia(t): the phase of arrival process at time t,
(N(t),J(t),I
s(t),I
a(t):t≥0}is a LIQBD with state space
Ω={{(0,j)/1≤j≤m}{(q,p,i,j)/q ≥1,1≤p≤k2,1≤i≤n, 1≤j≤m}}
https://doi.org/10.17993/3cemp.2022.110250.116-137
The infinitesimal generator of this CTMC is
Q∗=
B1B0
B2A1A0
A2A1A0
.........
.
Here
B1
is an
m×m
matrix which contains the transition within the level 0;
B0
is an
m×k2mn
matrix
which contains transitions from level 0to level 1;
B2
is a
k2mn ×m
matrix that contains transitions
from level 1to level 0;
A0
represents transitions from level
q
to level
q
+1for
q≥
1,
A1
represents
transitions within the level
q
for
q≥
1and
A2
represents transitions from level
q
to
q−
1for
q≥
2. All
these are square matrices of order k2mn ×k2mn.
B1=D0
B0=β
β
β⊗D10
0
0
B2=
S0⊗Im
S0⊗Im
S0⊗Im
.
.
.
(S0+enθk2)⊗Im
A1=
C1Imnθ1
C2Imnθ1
C3Imnθ3
......
Ck1Imnθk1
.........
Ck2−1Imnθk2−1
Ck2
where Ch=S⊗Im+In⊗D0−Imnθh,1≤h≤k2
A2=
S0β
β
β⊗Im0
0
00
0
00
0
0
S0β
β
β⊗Im0
0
00
0
00
0
0
S0β
β
β⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
(S0+enθk2)β
β
β⊗Im0
0
00
0
00
0
0
A0=
In⊗D1
In⊗D1
In⊗D1
.........
In⊗D1
3 Steady State Analysis
In this section, we perform the steady state analysis of the queueing model under study by first
establishing the stability condition of the queueing system.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
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120
Notations and abbreviations used
Notations and abbreviations used
Notations and abbreviations used
•LIQBD: Level independent Quasi-Birth and Death.
•MAP : Markovian Arrival Process.
•CTMC: Continuous time Markov chain.
•IP: Identity matrix of order P.
•ea
ea
ea: Column vector of 1′s of order a.
•e
e
e: Column vector of 1′s of appropriate order.
•x′
x′
x′: Transpose of a vector x
x
x.
The remaining part of this paper is organized as follows. In section 2 the model under study is
mathematically formulated. In section 3 we perform the steady-state analysis of the queueing model.
Service time analysis and waiting time analysis of a customer are discussed in sections 4 and 5
respectively. Some additional performance measures are provided in section 6. A revenue function is
discussed in section 7. Model description and mathematical formulation of model 2 are given in section
8. In section 9 we perform the steady state analysis of model 2. Numerical results are discussed in
section 10.
2 Mathematical formulation of Model I
We consider a single-server queueing system in which the arrival process follows MAP with representation
D
=(
D0,D
1
)of order m and service time follows continuous phase-type distribution (
β
β
β,S
)of order n.
When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has
k
stages. The
pth
stage parameter is
θp
for 1
≤p≤k
. If a customer completes the service in between
the realizations of stages
k1
and
k2
(1
<k
1<k
2<k
) of the clock, it is perfect. If a customer completes
the service either before the
kth
1
stage realization or after the
kth
2
stage realization, it is discarded. The
expected service rate is
µ
=[
β
β
β
(
−S
)
−1e
]
−1
. Let
D
=
D0
+
D1
be the infinitesimal generator matrix of
the arrival process and
δ
δ
δ
be its stationary probability vector, then
δ
δ
δD
=0
,δ
δ
δe
=1
.
The constant
λ
=
δ
δ
δD1ereferred to as the fundamental rate, gives the expected number of arrivals per unit of time.
2.1 The QBD process
The model described in section 1 can be studied as a LIQBD process. First, we define the following
notations:
N(t): number of customers in the system at time t,
J(t)=j, if the Erlang clock is in the jth stage at time t,j=1,2, ..., k2,
Is(t): the phase of service process at time t,
Ia(t): the phase of arrival process at time t,
(N(t),J(t),I
s(t),I
a(t):t≥0}is a LIQBD with state space
Ω={{(0,j)/1≤j≤m}{(q,p,i,j)/q ≥1,1≤p≤k2,1≤i≤n, 1≤j≤m}}
https://doi.org/10.17993/3cemp.2022.110250.116-137
The infinitesimal generator of this CTMC is
Q∗=
B1B0
B2A1A0
A2A1A0
.........
.
Here
B1
is an
m×m
matrix which contains the transition within the level 0;
B0
is an
m×k2mn
matrix
which contains transitions from level 0to level 1;
B2
is a
k2mn ×m
matrix that contains transitions
from level 1to level 0;
A0
represents transitions from level
q
to level
q
+1for
q≥
1,
A1
represents
transitions within the level
q
for
q≥
1and
A2
represents transitions from level
q
to
q−
1for
q≥
2. All
these are square matrices of order k2mn ×k2mn.
B1=D0
B0=β
β
β⊗D10
0
0
B2=
S0⊗Im
S0⊗Im
S0⊗Im
.
.
.
(S0+enθk2)⊗Im
A1=
C1Imnθ1
C2Imnθ1
C3Imnθ3
......
Ck1Imnθk1
.........
Ck2−1Imnθk2−1
Ck2
where Ch=S⊗Im+In⊗D0−Imnθh,1≤h≤k2
A2=
S0β
β
β⊗Im0
0
00
0
00
0
0
S0β
β
β⊗Im0
0
00
0
00
0
0
S0β
β
β⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
(S0+enθk2)β
β
β⊗Im0
0
00
0
00
0
0
A0=
In⊗D1
In⊗D1
In⊗D1
.........
In⊗D1
3 Steady State Analysis
In this section, we perform the steady state analysis of the queueing model under study by first
establishing the stability condition of the queueing system.
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3.1 Stability Condition
The generator matrix A=A0+A1+A2
A
=
In⊗D1+C1+S0β
β
β⊗ImImnθ1
S0β
β
β⊗ImIn⊗D1+C2Imnθ2
S0β
β
β⊗Im0
0
0In⊗D1+C3Imnθ3
......
S0β
β
β⊗ImIn⊗D1+Ck1Imnθk1
......
(S0β
β
β+eθk2β)
β)
β)⊗ImIn⊗D1+Ck2
.
Let
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
denote the steady state probability vector of the generator matrix
A.
Here O(π
π
π)=1×k2mn and the O(πr
πr
πr)=1×nm for 1≤r≤k2.
Steady state probability vector πsatisfying the equations
.
πA
πA
πA =0,π
π
πe=1.(1)
Using equation (1), we get
π1
π1
π1[In⊗D1+C1+S0β
β
β⊗Im]+(π2
π2
π2+π3
π3
π3+π4
π4
π4+.....+πk1
πk1
πk1+.....+πk2−1
πk2−1
πk2−1)[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0β
β
β+eθk2β)
β)
β)⊗Im]=0
0
0(2)
π1
π1
π1Imnθ1+π2
π2
π2[In⊗D1+C2]=0
0
0(3)
π2
π2
π2Imnθ2+π3
π3
π3[In⊗D1+C3]=0
0
0(4)
π3
π3
π3Imnθ3+π4
π4
π4[In⊗D1+C4]=0
0
0(5)
πk1
πk1
πk1Imnθk1+πk1+1
πk1+1
πk1+1[In⊗D1+Ck1+1]=0
0
0(6)
πk2−1
πk2−1
πk2−1Imnθk2−1+πk2
πk2
πk2[In⊗D1+Ck2]=0
0
0(7)
π1
π1
π1×e+π2
π2
π2×e+......... +πk1
πk1
πk1×e+........ ++πk2
πk2
πk2×e=1 (8)
From equation (7);
πk2−1
πk2−1
πk2−1=−πk2
πk2
πk2[In⊗D1+Ck2]1
θk2−1
Imn (9)
By back substitution and using equation (8) we get all the values of
πr
πr
πr′s
. Thus we get the steady-state
probability vector of A.
The
LIQBD
description of the model indicates that the queueing system is stable if and only if the
left drift exceeds that of the right drift. That is,
π
π
πA0e<π
π
πA2e.(10)
π
π
πA0e=π1
π1
π1(In⊗D1)e+π2
π2
π2(In⊗D1)e+........ +πk1
πk1
πk1(In⊗D1)e+....... +πk2
πk2
πk2(In⊗D1)e=
k2
r=1
πr(In⊗D1)e(11)
π
π
πA2e=π1
π1
π1[S0β
β
β⊗Im]e+π2
π2
π2[S0β
β
β⊗Im]e+....... +πk1
πk1
πk1[S0β
β
β⊗Im]e+........ +πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e
=
k2−1
r=1
πr
πr
πr[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e(12)
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Therefore the stability condition is
k2
r=1
πr
πr
πr(In⊗D1)e<
k2−1
r=1
πr
πr
πr[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e(13)
3.2 The Steady State Probability Vector of Q
Let x
x
xbe the steady state probability vector of Q.
x
x
x
=(
x0,x
1,x
2
x0,x
1,x
2
x0,x
1,x
2...
)
,
where
x0
x0
x0
is of dimension 1
×m
and
x1,x
2
x1,x
2
x1,x
2,...
are each of dimension 1
×k2mn
.
Under the stability condition, we have
xi
xi
xi
=
x
x
x1Ri−1,i ≥
2, where the matrix
R
is the minimal
nonnegative solution to the matrix quadratic equation
R2A2+RA1+A0=0
and the vectors x0
x0
x0and x1
x1
x1are obtained by solving the equations
x0
x0
x0B1+x1
x1
x1B2=0 (14)
x0
x0
x0B0+x1
x1
x1(A1+RA2)=0 (15)
subject to the normalizing condition
x0
x0
x0e
e
e+x1
x1
x1(I−R)−1e
e
e=1 (16)
Solving equations (15),(16) and(17), we get x0
x0
x0and x1
x1
x1. Hence we can find all xi
xi
xi’s.
4 Analysis of Service Time of a Customer
We consider a Markov Process Y(t)={(J(t),I
s(t)) : t≥0}where
J(t)=j, if the Erlang clock is in the jth stage at time t,j=1,2, ..., k2.
Is(t): the phase of service process at time t
The state space of this process is
Ω
1
=
{1,2, ..k1, ...k2}×{1,2,3, ..., n}{
∆
1}{
∆
2}
,where ∆
1
and ∆
2
denote the absorbing states.
∆
1
denotes the absorption occur due to service completion and ∆
2
denotes absorption occur due to
realization of kth
2stage of the Erlang clock.
The infinitesimal generator matrix is
Q1=
S−θ1Iθ
1IS
00
0
0
S−θ2Iθ
2IS
00
0
0
......
S−θk1Iθ
k1I
......
S−θk2IS
0eθk2
.
where S1=
S−θ1Iθ
1I
S−θ2Iθ
2I......
S−θk1Iθ
k1I......
S−θk2I
.
The initial probability vector is α
α
α=(β
β
β,0
0
0,0
0
0, ....,0
0
0)
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
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3.1 Stability Condition
The generator matrix A=A0+A1+A2
A
=
In⊗D1+C1+S0β
β
β⊗ImImnθ1
S0β
β
β⊗ImIn⊗D1+C2Imnθ2
S0β
β
β⊗Im0
0
0In⊗D1+C3Imnθ3
......
S0β
β
β⊗ImIn⊗D1+Ck1Imnθk1
......
(S0β
β
β+eθk2β)
β)
β)⊗ImIn⊗D1+Ck2
.
Let
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
denote the steady state probability vector of the generator matrix
A.
Here O(π
π
π)=1×k2mn and the O(πr
πr
πr)=1×nm for 1≤r≤k2.
Steady state probability vector πsatisfying the equations
.
πA
πA
πA =0,π
π
πe=1.(1)
Using equation (1), we get
π1
π1
π1[In⊗D1+C1+S0β
β
β⊗Im]+(π2
π2
π2+π3
π3
π3+π4
π4
π4+.....+πk1
πk1
πk1+.....+πk2−1
πk2−1
πk2−1)[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0β
β
β+eθk2β)
β)
β)⊗Im]=0
0
0(2)
π1
π1
π1Imnθ1+π2
π2
π2[In⊗D1+C2]=0
0
0(3)
π2
π2
π2Imnθ2+π3
π3
π3[In⊗D1+C3]=0
0
0(4)
π3
π3
π3Imnθ3+π4
π4
π4[In⊗D1+C4]=0
0
0(5)
πk1
πk1
πk1Imnθk1+πk1+1
πk1+1
πk1+1[In⊗D1+Ck1+1]=0
0
0(6)
πk2−1
πk2−1
πk2−1Imnθk2−1+πk2
πk2
πk2[In⊗D1+Ck2]=0
0
0(7)
π1
π1
π1×e+π2
π2
π2×e+......... +πk1
πk1
πk1×e+........ ++πk2
πk2
πk2×e=1 (8)
From equation (7);
πk2−1
πk2−1
πk2−1=−πk2
πk2
πk2[In⊗D1+Ck2]1
θk2−1
Imn (9)
By back substitution and using equation (8) we get all the values of
πr
πr
πr′s
. Thus we get the steady-state
probability vector of A.
The
LIQBD
description of the model indicates that the queueing system is stable if and only if the
left drift exceeds that of the right drift. That is,
π
π
πA0e<π
π
πA2e.(10)
π
π
πA0e=π1
π1
π1(In⊗D1)e+π2
π2
π2(In⊗D1)e+........ +πk1
πk1
πk1(In⊗D1)e+....... +πk2
πk2
πk2(In⊗D1)e=
k2
r=1
πr(In⊗D1)e(11)
π
π
πA2e=π1
π1
π1[S0β
β
β⊗Im]e+π2
π2
π2[S0β
β
β⊗Im]e+....... +πk1
πk1
πk1[S0β
β
β⊗Im]e+........ +πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e
=
k2−1
r=1
πr
πr
πr[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e(12)
https://doi.org/10.17993/3cemp.2022.110250.116-137
Therefore the stability condition is
k2
r=1
πr
πr
πr(In⊗D1)e<
k2−1
r=1
πr
πr
πr[S0β
β
β⊗Im]+πk2
πk2
πk2[(S0+enθk2)β
β
β⊗Im]e(13)
3.2 The Steady State Probability Vector of Q
Let x
x
xbe the steady state probability vector of Q.
x
x
x
=(
x0,x
1,x
2
x0,x
1,x
2
x0,x
1,x
2...
)
,
where
x0
x0
x0
is of dimension 1
×m
and
x1,x
2
x1,x
2
x1,x
2,...
are each of dimension 1
×k2mn
.
Under the stability condition, we have
xi
xi
xi
=
x
x
x1Ri−1,i ≥
2, where the matrix
R
is the minimal
nonnegative solution to the matrix quadratic equation
R2A2+RA1+A0=0
and the vectors x0
x0
x0and x1
x1
x1are obtained by solving the equations
x0
x0
x0B1+x1
x1
x1B2=0 (14)
x0
x0
x0B0+x1
x1
x1(A1+RA2)=0 (15)
subject to the normalizing condition
x0
x0
x0e
e
e+x1
x1
x1(I−R)−1e
e
e=1 (16)
Solving equations (15),(16) and(17), we get x0
x0
x0and x1
x1
x1. Hence we can find all xi
xi
xi’s.
4 Analysis of Service Time of a Customer
We consider a Markov Process Y(t)={(J(t),I
s(t)) : t≥0}where
J(t)=j, if the Erlang clock is in the jth stage at time t,j=1,2, ..., k2.
Is(t): the phase of service process at time t
The state space of this process is
Ω
1
=
{1,2, ..k1, ...k2}×{1,2,3, ..., n}{
∆
1}{
∆
2}
,where ∆
1
and ∆
2
denote the absorbing states.
∆
1
denotes the absorption occur due to service completion and ∆
2
denotes absorption occur due to
realization of kth
2stage of the Erlang clock.
The infinitesimal generator matrix is
Q1=
S−θ1Iθ
1
IS
00
0
0
S−θ2Iθ
2
IS
00
0
0
......
S−θk1Iθ
k1I
......
S−θk2IS
0eθk2
.
where S1=
S−θ1Iθ
1I
S−θ2Iθ
2I......
S−θk1Iθ
k1I......
S−θk2I
.
The initial probability vector is α
α
α=(β
β
β,0
0
0,0
0
0, ....,0
0
0)
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The expected service time of a customer is the time until absorption of the above process
which is given by ES =α
α
α(−S−1
1)e
5 Waiting Time Analysis
To find the expected waiting time of a tagged customer who joins as the
r
th customer in system, we
consider the Markov Processes
W={W(t):t≥0}={(N(t),J(t),I
s(t)) : t≥0}where
N(t)-Rank of the customer in the system at time t
J(t)=j, if the Erlang clockis in the jth stage at time t,j=1,2, ..., k2.
Is(t)- Phase of the service at time t
The rank of the customer decrease by one when a customer ahead of him completes the service. The
rank of the customer is assumed to be
r
if he joins as the
r
th customer in the system. State-space of
W(t)is Ω2={{r, r −1,r−2,···,2}×{1,2,3, ...k2}×{1,2,3, ...., n}} ∪ {∆∗}
where ∆
∗
denotes the absorbing state. That is ∆
∗
denotes the state that the tagged customer selected
for service.
The infinitesimal generator is
W=
0
0
00
0
0
T0T
T0β
β
βT
T0β
β
βT
......
.
where T=
S−θ1Iθ
1I
S−θ2Iθ
2I......
S−θk1Iθ
k1I......
S−θk2I
.
T0=
S00
0
00
0
0
S00
0
00
0
0
S00
0
00
0
0
.
.
.
S0+eθk20
0
00
0
0
.
Let
yrpi
be the steady-state probability that an arriving customer finds the server in busy with
current service phase
i
, Erlang clock is in
pth
level and the number of customers in the system including
the current arrival tobe rfor 1≤p≤k2and 1≤i≤n
Let yr=(yr11,y
r12, , .....yr1n,y
r21,y
r22, .....yr2n, ....., yrk21,y
rk22, .....yrk2n)
and y= (0,y
2,y
3.....yr)
https://doi.org/10.17993/3cemp.2022.110250.116-137
Then yr=xr−1(I⊗D1
λ),r ≥2
Waiting time is the time until absorption of the Markov chain is given by Ω
2
. Let
W
(
s
)denote the
Laplace Stieltjes Transform (LST) of waiting time in the queue of an arrival.
Theorem 1. The LST of the waiting time distribution of an arriving customer is
W
(
s
)=
c∞
r=2 yr
(
sI −T
)
−1T0
[
β
β
β
(
sI −T
)
−1T0
]
r−2, Re
(
s
)
≥
0, where the normalising constant c is given by
c=[
∞
r=2 yre]−1
6 Additional Performance Measures
•probability that the system is empty:
P0=x0
x0
x0e.
•Probability that qcustomers in the system:
Pq=xq
xq
xqe.
•Probability that the server is busy:
Pbusy =
∞
q=1
k2
p=1
n
i=1
m
j=1
xqpij
xqpij
xqpij .
•Expected number of customers in the queue:
ECQ =
∞
q=1
(q−1)xqe
xqe
xqe.
•Expected number of customers in the system:
ECS =
∞
q=0
qxqe
xqe
xqe.
.
•Rate at which customers discarded before kth
1stage realization of Erlang clock
RK1=
∞
q=1
k1
p=1
n
i=1
m
j=1
xqpij
xqpij
xqpij S0e.
•Rate at which customers discard after kth
2stage realization of Erlang clock
RK2=
∞
q=1
n
i=1
m
j=1
xqk2ij
xqk2ij
xqk2ij θk2.
•Rate at which customers depart with successful completion of service
RP =
∞
q=1
k2
p=k1+1
n
i=1
m
j=1
xqpij
xqpij
xqpij S0e.
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
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124
The expected service time of a customer is the time until absorption of the above process
which is given by ES =α
α
α(−S−1
1)e
5 Waiting Time Analysis
To find the expected waiting time of a tagged customer who joins as the
r
th customer in system, we
consider the Markov Processes
W={W(t):t≥0}={(N(t),J(t),I
s(t)) : t≥0}where
N(t)-Rank of the customer in the system at time t
J(t)=j, if the Erlang clockis in the jth stage at time t,j=1,2, ..., k2.
Is(t)- Phase of the service at time t
The rank of the customer decrease by one when a customer ahead of him completes the service. The
rank of the customer is assumed to be
r
if he joins as the
r
th customer in the system. State-space of
W(t)is Ω2={{r, r −1,r−2,···,2}×{1,2,3, ...k2}×{1,2,3, ...., n}} ∪ {∆∗}
where ∆
∗
denotes the absorbing state. That is ∆
∗
denotes the state that the tagged customer selected
for service.
The infinitesimal generator is
W=
0
0
00
0
0
T0T
T0β
β
βT
T0β
β
βT
......
.
where T=
S−θ1Iθ
1I
S−θ2Iθ
2I......
S−θk1Iθ
k1I......
S−θk2I
.
T0=
S00
0
00
0
0
S00
0
00
0
0
S00
0
00
0
0
.
.
.
S0+eθk20
0
00
0
0
.
Let
yrpi
be the steady-state probability that an arriving customer finds the server in busy with
current service phase
i
, Erlang clock is in
pth
level and the number of customers in the system including
the current arrival tobe rfor 1≤p≤k2and 1≤i≤n
Let yr=(yr11,y
r12, , .....yr1n,y
r21,y
r22, .....yr2n, ....., yrk21,y
rk22, .....yrk2n)
and y= (0,y
2,y
3.....yr)
https://doi.org/10.17993/3cemp.2022.110250.116-137
Then yr=xr−1(I⊗D1
λ),r ≥2
Waiting time is the time until absorption of the Markov chain is given by Ω
2
. Let
W
(
s
)denote the
Laplace Stieltjes Transform (LST) of waiting time in the queue of an arrival.
Theorem 1. The LST of the waiting time distribution of an arriving customer is
W
(
s
)=
c∞
r=2 yr
(
sI −T
)
−1T0
[
β
β
β
(
sI −T
)
−1T0
]
r−2, Re
(
s
)
≥
0, where the normalising constant c is given by
c=[
∞
r=2 yre]−1
6 Additional Performance Measures
•probability that the system is empty:
P0=x0
x0
x0e.
•Probability that qcustomers in the system:
Pq=xq
xq
xqe.
•Probability that the server is busy:
Pbusy =
∞
q=1
k2
p=1
n
i=1
m
j=1
xqpij
xqpij
xqpij .
•Expected number of customers in the queue:
ECQ =
∞
q=1
(q−1)xqe
xqe
xqe.
•Expected number of customers in the system:
ECS =
∞
q=0
qxqe
xqe
xqe.
.
•Rate at which customers discarded before kth
1stage realization of Erlang clock
RK1=
∞
q=1
k1
p=1
n
i=1
m
j=1
xqpij
xqpij
xqpij S0e.
•Rate at which customers discard after kth
2stage realization of Erlang clock
RK2=
∞
q=1
n
i=1
m
j=1
xqk2ij
xqk2ij
xqk2ij θk2.
•Rate at which customers depart with successful completion of service
RP =
∞
q=1
k2
p=k1+1
n
i=1
m
j=1
xqpij
xqpij
xqpij S0e.
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7 Revenue Function
Based on the above performance measures, we construct a revenue function as follows.
CK1
- Unit time cost of service when customer discarded before the
kth
1
stage realization of Erlang Clock.
CK2- Unit time cost of service when customer discarded after kth
2stage realization of Erlang clock.
RS - Revenue per unit time for successful service.
Then the expected revenue per unit time, ER =RP ×RS −RK1×CK1−RK2×CK2.
In this model, customers are discarded when either their service completes before reachig the stage
k1
or goes beyond the stage
k2
. To minimise the rate of discarding customers before reaching stage
k1
, we have to slow down the service rate up to
kth
1
stage realization of Erlang clock so as to get the
service cross the stage
k1
. Similarly to minimse the rate of discarding customers after
kth
2
stage, we
have to increase the service rate beyond
kth
1
stage realization of the Erlang clock to get the service
completed before crossing the boundary
k2
. Accordingly, we can reduce the loss to the system due
to imperfect service. The extra cost involved while increasing the service rate beyond
k1
gets com-
pensated through slow down of service rate up to the stage
k1
, and also through reduced imperfect service.
Next, we proceed to the analysis of Model II.
8 Model II
8.1 Model description and Mathematical Formulation
We consider a single server queueing system in which all assumptions are exactly same as in Model I
except the assumption on service time. Upto the stage
k1
service time follows phase-type distribution
(
α′
α′
α′,T′
)of order
n1
and beyond the stage
k1
, the service time follows phase-type distribution (
β′
β′
β′,S′
)of
order
n2
. Therefore the entire service time follows phase-type distribution (
γ′
γ′
γ′,L
)of order
n
=
n1
+
n2
,
which is the convolution of the two phase-type distributions (
α′
α′
α′,T′
)of order
n1
and (
β′
β′
β′,S′
)of order
n2
.
Then γ′=(α′
α′
α′,α
′
n1+1β′
β′
β′)=(α′
α′
α′,0
0
0),L=T′T′0β′
β′
β′
0
0
0S′.
Here we take α′
n1+1 =0and β′
n2+1 =0
The above described model can be studied as a LIQBD process.
Let
N(t): Number of customers in the system at time t,
J(t)=p, if the Erlang clock is in the pth stage at time t,p=1,2, ..., k2,
Is(t): the phase of service process at time t,
Ia(t): the phase of arrival process at time t.
(N(t),J(t),I
s(t),I
a(t):t≥0}is a LIQBD with state space
Ω
3
=
{{
(0
,j
)
/
1
≤j≤m}{
(
q, p, i, j
)
/q ≥
1
,
1
≤p≤k1,
1
≤i≤n1,
1
≤j≤m}{
(
q,p,i,j
)
/q ≥
1,(k1+ 1) ≤p≤k2,(n1+ 1) ≤i≤(n1+n2),1≤j≤m}}
https://doi.org/10.17993/3cemp.2022.110250.116-137
The infinitesimal generator of this CTMC is
Q∗=
B′
1B′
0
B′
2A′
1A′
0
A′
2A′
1A′
0
.........
.
Here
B′
1
is an
m×m
matrix that contains the transition within the level 0;
B′
0
is an
m×
[
k1n1m
+(
k2−
k1
)
mn2
]matrix which contains transitions from level 0to level 1;
B′
2
is a [
k1n1m
+(
k2−k1
)
mn2
]
×m
matrix which contains transitions from level 1to level 0;
A′
0
represents transitions from level
q
to
q
+1
for
q≥
1,
A′
1
represents transitions within the level
q
for
q≥
1and
A′
2
represents transitions from level
qto level q−1for q≥2. All these are square matrices of order [k1n1m+(k2−k1)mn2].
B1=B′
1=D0
B′
0=α′
α′
α′⊗D10
0
0
B′
2=
T′0⊗Im
T′0⊗Im
.
.
.
T′0⊗Im
S′0⊗Im
S′0⊗Im
.
.
.
S′0⊗Im
(S′0+en2θk2)⊗Im
A′
1=
F1Imn1θ1
F2Imn1θ1
......
Fk1θk1β′
β′
β′⊗Im
Ek1+1 Imn2θk1+2
Ek1+2 Imn2θk1+1
......
Ek2
where Ft=T′⊗Im+In1⊗D0−Imn1θt,1≤t≤k1
Er=S′⊗Im+In2⊗D0−Imn2θr,k
1+1≤r≤k2
A′
2=
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
S′0α′
α′
α′⊗Im0
0
00
0
00
0
0
S′0α′
α′
α′⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
S′0α′
α′
α′+en2θk2α′
α′
α′⊗Im0
0
00
0
00
0
0
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
126
7 Revenue Function
Based on the above performance measures, we construct a revenue function as follows.
CK1
- Unit time cost of service when customer discarded before the
kth
1
stage realization of Erlang Clock.
CK2- Unit time cost of service when customer discarded after kth
2stage realization of Erlang clock.
RS - Revenue per unit time for successful service.
Then the expected revenue per unit time, ER =RP ×RS −RK1×CK1−RK2×CK2.
In this model, customers are discarded when either their service completes before reachig the stage
k1
or goes beyond the stage
k2
. To minimise the rate of discarding customers before reaching stage
k1
, we have to slow down the service rate up to
kth
1
stage realization of Erlang clock so as to get the
service cross the stage
k1
. Similarly to minimse the rate of discarding customers after
kth
2
stage, we
have to increase the service rate beyond
kth
1
stage realization of the Erlang clock to get the service
completed before crossing the boundary
k2
. Accordingly, we can reduce the loss to the system due
to imperfect service. The extra cost involved while increasing the service rate beyond
k1
gets com-
pensated through slow down of service rate up to the stage
k1
, and also through reduced imperfect service.
Next, we proceed to the analysis of Model II.
8 Model II
8.1 Model description and Mathematical Formulation
We consider a single server queueing system in which all assumptions are exactly same as in Model I
except the assumption on service time. Upto the stage
k1
service time follows phase-type distribution
(
α′
α′
α′,T′
)of order
n1
and beyond the stage
k1
, the service time follows phase-type distribution (
β′
β′
β′,S′
)of
order
n2
. Therefore the entire service time follows phase-type distribution (
γ′
γ′
γ′,L
)of order
n
=
n1
+
n2
,
which is the convolution of the two phase-type distributions (
α′
α′
α′,T′
)of order
n1
and (
β′
β′
β′,S′
)of order
n2
.
Then γ′=(α′
α′
α′,α
′
n1+1β′
β′
β′)=(α′
α′
α′,0
0
0),L=T′T′0β′
β′
β′
0
0
0S′.
Here we take α′
n1+1 =0and β′
n2+1 =0
The above described model can be studied as a LIQBD process.
Let
N(t): Number of customers in the system at time t,
J(t)=p, if the Erlang clock is in the pth stage at time t,p=1,2, ..., k2,
Is(t): the phase of service process at time t,
Ia(t): the phase of arrival process at time t.
(N(t),J(t),I
s(t),I
a(t):t≥0}is a LIQBD with state space
Ω
3
=
{{
(0
,j
)
/
1
≤j≤m}{
(
q, p, i, j
)
/q ≥
1
,
1
≤p≤k1,
1
≤i≤n1,
1
≤j≤m}{
(
q,p,i,j
)
/q ≥
1,(k1+ 1) ≤p≤k2,(n1+ 1) ≤i≤(n1+n2),1≤j≤m}}
https://doi.org/10.17993/3cemp.2022.110250.116-137
The infinitesimal generator of this CTMC is
Q∗=
B′
1B′
0
B′
2A′
1A′
0
A′
2A′
1A′
0
.........
.
Here
B′
1
is an
m×m
matrix that contains the transition within the level 0;
B′
0
is an
m×
[
k1n1m
+(
k2−
k1
)
mn2
]matrix which contains transitions from level 0to level 1;
B′
2
is a [
k1n1m
+(
k2−k1
)
mn2
]
×m
matrix which contains transitions from level 1to level 0;
A′
0
represents transitions from level
q
to
q
+1
for
q≥
1,
A′
1
represents transitions within the level
q
for
q≥
1and
A′
2
represents transitions from level
qto level q−1for q≥2. All these are square matrices of order [k1n1m+(k2−k1)mn2].
B1=B′
1=D0
B′
0=α′
α′
α′⊗D10
0
0
B′
2=
T′0⊗Im
T′0⊗Im
.
.
.
T′0⊗Im
S′0⊗Im
S′0⊗Im
.
.
.
S′0⊗Im
(S′0+en2θk2)⊗Im
A′
1=
F1Imn1θ1
F2Imn1θ1
......
Fk1θk1β′
β′
β′⊗Im
Ek1+1 Imn2θk1+2
Ek1+2 Imn2θk1+1
......
Ek2
where Ft=T′⊗Im+In1⊗D0−Imn1θt,1≤t≤k1
Er=S′⊗Im+In2⊗D0−Imn2θr,k
1+1≤r≤k2
A′
2=
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
T′0α′
α′
α′⊗Im0
0
00
0
00
0
0
S′0α′
α′
α′⊗Im0
0
00
0
00
0
0
S′0α′
α′
α′⊗Im0
0
00
0
00
0
0
.
.
..
.
..
.
.
S′0α′
α′
α′+en2θk2α′
α′
α′⊗Im0
0
00
0
00
0
0
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A′
0=
In1⊗D1
In1⊗D1
......
In1⊗D1
In2⊗D1
......
In2⊗D1
=Ik1⊗(In1⊗D1)0
0
0
0
0
0I(k2−k1)⊗(In2⊗D1)
9 Steady State Analysis
In this section, we perform the steady state analysis of the queueing model under study by first
establishing the stability condition of the queueing system.
9.1 Stability Condition
The generator matrix A′=A′
0+A′
1+A′
2
A′=
In1⊗D1+F1+T′0α′
α′
α′⊗ImImn1θ1
T′0α′
α′
α′⊗ImIn1⊗D1+F2Imn1θ2
......
T′0α′
α′
α′⊗ImIn1⊗D1+Fk1θk1β′
β′
β′⊗Im
S′0α′
α′
α′⊗ImIn2⊗D1+Ek1+1 Imn2θk1+1
......
[(S′0+en2θk2)α′
α′
α′]⊗ImIn2⊗D1+Ek2
.
Let
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
denote the steady state probability vector of the generator matrix
A
Here
O
(
π
π
π
)=1
×
[
k1n1m
+(
k2−k1
)
n2m
]and the
O
(
πr
πr
πr
)=1
×n1m
for 1
≤r≤k1
and
O
(
πr
πr
πr
)=1
×n2m
for k1+1 ≤r≤k2.
Steady state probability vector πsatisfying the equations
.
πA′
πA′
πA′=0,π
π
πe=1.(17)
Using equation (18), we get
π1
π1
π1[In1⊗D1+F1+T′0α′
α′
α′⊗Im]+(π2
π2
π2+π3
π3
π3+π4
π4
π4+..... +πk1
πk1
πk1)[T′0α′
α′
α′⊗Im]
+(πk1+1
πk1+1
πk1+1 +..... +πk2−1
πk2−1
πk2−1)S′0α′
α′
α′⊗Im+πk2(S′0+en2θk2)α′
α′
α′⊗Im)=0
0
0
(18)
π1
π1
π1Imn1θ1+π2
π2
π2[In1⊗D1+F2]=0
0
0(19)
π2
π2
π2Imn1θ2+π3
π3
π3[In1⊗D1+F3]=0
0
0(20)
π3
π3
π3Imn1θ3+π4
π4
π4[In1⊗D1+F4]=0
0
0(21)
https://doi.org/10.17993/3cemp.2022.110250.116-137
πk1−1
πk1−1
πk1−1Imn2θk1−1+πk1
πk1
πk1[In1⊗D1+Fk1]=0
0
0(22)
πk1
πk1
πk1(θk1β′
β′
β′⊗Im)+πk1+1
πk1+1
πk1+1[In2⊗D1+Ek1+1]=0
0
0(23)
πk2−2
πk2−2
πk2−2Imn2θk2−2+πk2−1
πk2−1
πk2−1[In2⊗D1+Ek2−1]=0
0
0(24)
πk2−1
πk2−1
πk2−1Imn2θk2−1+πk2
πk2
πk2[In2⊗D1+Ek2]=0
0
0(25)
π1
π1
π1×e+π2
π2
π2×e+......... +πk1
πk1
πk1×e+........ ++πk2
πk2
πk2×e=1 (26)
From equation (25);
πk2−1
πk2−1
πk2−1=−πk2
πk2
πk2[In2⊗D1+Ek2]1
θk2−1
Imn2(27)
By back substitution and using equation (26) we get all the values of
πr
πr
πr′s
.Thus we get the steady-state
probability vector of A′.
The
LIQBD
description of the model indicates that the queueing system is stable if and only if the
left drift exceeds that of the right drift. That is,
π
π
πA′
0e<π
π
πA′
2e.(28)
Therefore the stability condition is
k1
r=1
πr
πr
πr(In1⊗D1)e+
k2
r=k1+1
πr
πr
πr(In2⊗D1)e<
k1
r=1
πr
πr
πr[T′0α′
α′
α′⊗Im]e+
k2
r=k1+1
πr
πr
πr(S′0α′
α′
α′⊗Im)e+πk2
πk2
πk2(en2θk2α′
α′
α′⊗Im)e
(29)
9.2 The Steady State Probability Vector of Q
Let x
x
xbe the steady state probability vector of Q.
x
x
x
=(
x0,x
1,x
2
x0,x
1,x
2
x0,x
1,x
2...
)
,
where
x0
x0
x0
is of dimension 1
×m
and
x1,x
2
x1,x
2
x1,x
2,...
are each of dimension 1
×
[k1mn1+(k2−k1)n2m].
Under the stability condition, we have
xi
xi
xi
=
x
x
x1Ri−1,i ≥
2, where the matrix
R
is the minimal
nonnegative solution to the matrix quadratic equation
R2A2+RA1+A0=0
and the vectors x0
x0
x0and x1
x1
x1are obtained by solving the equations
x0
x0
x0B1+x1
x1
x1B2=0 (30)
x0
x0
x0B0+x1
x1
x1(A1+RA2)=0 (31)
subject to the normalizing condition
x0
x0
x0e
e
e+x1
x1
x1(I−R)−1e
e
e=1 (32)
Solving equations (31), (32) and (33), we get x0
x0
x0and x1
x1
x1
Hence we can find all xi
xi
xi’s.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
128
A′
0=
In1⊗D1
In1⊗D1
......
In1⊗D1
In2⊗D1
......
In2⊗D1
=Ik1⊗(In1⊗D1)0
0
0
0
0
0I(k2−k1)⊗(In2⊗D1)
9 Steady State Analysis
In this section, we perform the steady state analysis of the queueing model under study by first
establishing the stability condition of the queueing system.
9.1 Stability Condition
The generator matrix A′=A′
0+A′
1+A′
2
A′=
In1⊗D1+F1+T′0α′
α′
α′⊗ImImn1θ1
T′0α′
α′
α′⊗ImIn1⊗D1+F2Imn1θ2
......
T′0α′
α′
α′⊗ImIn1⊗D1+Fk1θk1β′
β′
β′⊗Im
S′0α′
α′
α′⊗ImIn2⊗D1+Ek1+1 Imn2θk1+1
......
[(S′0+en2θk2)α′
α′
α′]⊗ImIn2⊗D1+Ek2
.
Let
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
π=(π1,π
2,π
3, .., πk1, ...., πk2)
denote the steady state probability vector of the generator matrix
A
Here
O
(
π
π
π
)=1
×
[
k1n1m
+(
k2−k1
)
n2m
]and the
O
(
πr
πr
πr
)=1
×n1m
for 1
≤r≤k1
and
O
(
πr
πr
πr
)=1
×n2m
for k1+1 ≤r≤k2.
Steady state probability vector πsatisfying the equations
.
πA′
πA′
πA′=0,π
π
πe=1.(17)
Using equation (18), we get
π1
π1
π1[In1⊗D1+F1+T′0α′
α′
α′⊗Im]+(π2
π2
π2+π3
π3
π3+π4
π4
π4+..... +πk1
πk1
πk1)[T′0α′
α′
α′⊗Im]
+(πk1+1
πk1+1
πk1+1 +..... +πk2−1
πk2−1
πk2−1)S′0α′
α′
α′⊗Im+πk2(S′0+en2θk2)α′
α′
α′⊗Im)=0
0
0
(18)
π1
π1
π1Imn1θ1+π2
π2
π2[In1⊗D1+F2]=0
0
0(19)
π2
π2
π2Imn1θ2+π3
π3
π3[In1⊗D1+F3]=0
0
0(20)
π3
π3
π3Imn1θ3+π4
π4
π4[In1⊗D1+F4]=0
0
0(21)
https://doi.org/10.17993/3cemp.2022.110250.116-137
πk1−1
πk1−1
πk1−1Imn2θk1−1+πk1
πk1
πk1[In1⊗D1+Fk1]=0
0
0(22)
πk1
πk1
πk1(θk1β′
β′
β′⊗Im)+πk1+1
πk1+1
πk1+1[In2⊗D1+Ek1+1]=0
0
0(23)
πk2−2
πk2−2
πk2−2Imn2θk2−2+πk2−1
πk2−1
πk2−1[In2⊗D1+Ek2−1]=0
0
0(24)
πk2−1
πk2−1
πk2−1Imn2θk2−1+πk2
πk2
πk2[In2⊗D1+Ek2]=0
0
0(25)
π1
π1
π1×e+π2
π2
π2×e+......... +πk1
πk1
πk1×e+........ ++πk2
πk2
πk2×e=1 (26)
From equation (25);
πk2−1
πk2−1
πk2−1=−πk2
πk2
πk2[In2⊗D1+Ek2]1
θk2−1
Imn2(27)
By back substitution and using equation (26) we get all the values of
πr
πr
πr′s
.Thus we get the steady-state
probability vector of A′.
The
LIQBD
description of the model indicates that the queueing system is stable if and only if the
left drift exceeds that of the right drift. That is,
π
π
πA′
0e<π
π
πA′
2e.(28)
Therefore the stability condition is
k1
r=1
πr
πr
πr(In1⊗D1)e+
k2
r=k1+1
πr
πr
πr(In2⊗D1)e<
k1
r=1
πr
πr
πr[T′0α′
α′
α′⊗Im]e+
k2
r=k1+1
πr
πr
πr(S′0α′
α′
α′⊗Im)e+πk2
πk2
πk2(en2θk2α′
α′
α′⊗Im)e
(29)
9.2 The Steady State Probability Vector of Q
Let x
x
xbe the steady state probability vector of Q.
x
x
x
=(
x0,x
1,x
2
x0,x
1,x
2
x0,x
1,x
2...
)
,
where
x0
x0
x0
is of dimension 1
×m
and
x1,x
2
x1,x
2
x1,x
2,...
are each of dimension 1
×
[k1mn1+(k2−k1)n2m].
Under the stability condition, we have
xi
xi
xi
=
x
x
x1Ri−1,i ≥
2, where the matrix
R
is the minimal
nonnegative solution to the matrix quadratic equation
R2A2+RA1+A0=0
and the vectors x0
x0
x0and x1
x1
x1are obtained by solving the equations
x0
x0
x0B1+x1
x1
x1B2=0 (30)
x0
x0
x0B0+x1
x1
x1(A1+RA2)=0 (31)
subject to the normalizing condition
x0
x0
x0e
e
e+x1
x1
x1(I−R)−1e
e
e=1 (32)
Solving equations (31), (32) and (33), we get x0
x0
x0and x1
x1
x1
Hence we can find all xi
xi
xi’s.
https://doi.org/10.17993/3cemp.2022.110250.116-137
129
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
10 Numerical Results
For the arrival process of customers, we consider the following three sets of matrices for D0and D1
MAP with positive correlation (MPC)
MAP with positive correlation (MPC)
MAP with positive correlation (MPC)
D0=
−1.7615 1.7615 0
0−1.7615 0
00−11.7054
,D1=
00 0
1.6294 0 0.1321
0.1233 0 11.5821
MAP with negative correlation (MNC)
MAP with negative correlation (MNC)
MAP with negative correlation (MNC)
D0=
−55 0
0−50
00−40.5
,D1=
000
0.1504.85
40.30 0.2
MAP with zero correlation (MZC)
MAP with zero correlation (MZC)
MAP with zero correlation (MZC)
D0=
−10 1
0−10
00−5.25
,D1=
000
0.9500.05
0.15 0 5.1
The arrival process labeled
MPC
has correlated arrivals with the correlation between two successive
interarrival times given by 0.5315, the arrival process corresponding to the one labeled
MNC
has
correlated arrivals with the correlation between two successive interarrival times given by -0.4470 and
the arrival process labeled MZC has zero correlation between two successive interarrival times.
Service time follows continuous phase-type distribution (β
β
β,S)of order 6 in Model I.
Here we take β
β
β= (0.2,0.1,0.2,0.1,0.2,0.2)
S=
−6.70.501.200.5
0−5.50.50.210.8
00.1−5.50.90.31.2
0.100.8−6.50.30
0.300.50−4.51.3
0.20.300.40−5.5
.
In Model II, α′
α′
α′= (0.2,0.5,0.3),β′
β′
β′= (0.1,0.4,0.5),γ′
γ′
γ′=(α′
α′
α′,0
0
0)=(0.2,0.5,0.3,0,0,0).
S′=
−28.19 0.50
0−28.21 0.1
00.2−28.46
,T′=
−4.5583 0.30
0−4.582 0.3
00.1−4.6
,
L=T′T′0β′
β′
β′
0
0
0S′=
−4.5583 0.3 00.4258 1.7033 2.1292
0−4.582 0.30.4282 1.7128 2.1410
00.1−4.60.45 1.82.25
0 00−28.1900 0.50
0 00 0−28.2100 0.1
0 00 0 0.2−28..46
.
Service rate in Model I = Service rate in Model II= 3.7649. Fix
n
=6
,n
1
=3
,n
2
=3
,m
=3
,k
1
=
2,k
2=6,k =7,CK1 = 12,CK2 = 20, RS = 40.
Let ER1and ER2denote the expected revenue in Model I and Model II respectively.
https://doi.org/10.17993/3cemp.2022.110250.116-137
10.1 MAP with positive correlation (MPC) and the clock follows generalized Erlang distribution
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 51.2838 50.6032 8.0993 0.6468 7.4011 185.9164
13-13.5 48.9982 48.3254 7.8161 0.7322 7.5104 191.9783
14-14.5 46.7988 46.1337 7.5605 0.8178 7.5929 196.6326
15-15.5 44.6870 44.0296 7.3284 0.9032 7.6535 200.1340
16-16.5 42.6620 42.0121 7.1163 0.9881 7.6960 202.6849
17-17.5 40.7213 40.0788 6.9215 1.0722 7.7237 204.4475
18-18.5 38.8617 38.2264 6.7416 1.1555 7.7390 205.5523
19-19.5 37.0791 36.4510 6.5748 1.2378 7.7439 206.1053
20-20.5 35.3697 34.7486 6.4194 1.3190 7.7401 206.1931
206.1931
206.1931
21-21.5 33.7293 33.1151 6.2741 1.3990 7.7289 205.8866
22-22.5 32.1541 31.5466 6.1377 1.4778 7.7113 205.2444
23-23.5 30.6401 30.0393 6.0093 1.5554 7.6883 204.3152
24-24.5 29.1838 28.5895 5.8879 1.6316 7.6604 203.1393
25-25.5 27.7819 27.1941 5.7727 1.7065 7.6288 201.7505
26-26.5 26.4132 25.8498 5.6633 1.7800 7.5934 200.1771
Table 1 – Effect of θi’s on performance measures in Model I when the arrival process is MPC
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 19.3578 18.8077 6.0942 0.0214 7.0839 209.7992
14-14.5 15.2272 14.7401 5.6427 0.0350 7.7778 242.6997
16-16.5 11.8777 11.3801 5.2328 0.0524 8.3303 269.3707
18-18.5 9.168- 8.6946 4.8598 0.0374 8.7588 290.5665
20-20.5 7.0021 6.5515 4.5207 0.0977 9.0821 307.0833
22-22.5 5.3114 4.8821 4.2149 0.1248 9.3219 319.7984
24-24.5 4.0354 3.6254 3.9425 0.1546 9.5012 329.6455
26-26.5 3.1050 2.7124 3.7025 0.1868 9.6401 337.4388
28-28.5 2.4421 2.0649 3.4912 0.2211 9.7500 343.6853
30-30.5 1.9721 1.6087 3.3036 0.2573 9.8350 348.6118
32-32.5 1.6352 1.2844 3.1352 0.2951 9.8966 352.3382
34-34.5 1.3886 1.0493 2.9829 0.3342 9.9367 354.9907
36-36.5 1.2035 0.8749 2.8443 0.3742 9.9582 356.7130
37-37.5 1.1279 0.8044 2.7796 0.3946 9.9629 357.2698
38-38.5 1.0611 0.7425 2.7177 0.4151 9.9640 357.6454
39-39.5 1.0019 0.6879 2.6585 0.4357 9.9618 357.8543
40-40.5 0.9490 0.6395 2.6018 0.4565 9.9565 357.9098
357.9098
357.9098
41-41.5 0.9015 0.5964 2.5474 0.4774 9.9485 357.8242
42-42.5 0.8587 0.5578 2.4952 0.4983 9.9379 357.6082
43-43.5 0.8200 0.5231 2.4451 0.5194 9.9250 357.2720
44-44.5 0.7847 0.4918 2.3970 0.5405 9.9100 356.8247
45-45.5 0.7525 0.4634 2.3507 0.5616 9.8929 356.2746
Table 2 – Effect of θi’s on performance measures in Model II when the arrival process is MPC
Table 1 and Table 2 show the effect of
θ′
is
on various performance measures and the revenue function
in Model I and II respecively when the arrival process is MPC. In Model I, when
θ′
is
values increases,
the values of
ER1
increase and reach the maximum at
θ′
i
s=19-19.5 and then decreases. The maximum
revenue in this case is 206.1931. In Model II, when
θ′
is
values increases, the values of
ER2
increase and
reach the maximum at
θ′
is
= 40
−
40
.
5, and then decreases. The maximum value of
ER2
is 357.9098.
When we compare Model I and II, the values expected revenue in Model II is greater than that of the
corresponding values of expected revenue in Model I. Also, the values of the rate of perfect service
(RP) in Model II are greater than the corresponding values in Model I. In both models values of
RK1
decreases when
θ′
is
values increases. Also in both models,
RK2
increases when
θ′
is
values increases. This
is because when
θ′
is
values increase, the expected service time of the customer in each stage decreases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
130
10 Numerical Results
For the arrival process of customers, we consider the following three sets of matrices for D0and D1
MAP with positive correlation (MPC)
MAP with positive correlation (MPC)
MAP with positive correlation (MPC)
D0=
−1.7615 1.7615 0
0−1.7615 0
00−11.7054
,D1=
00 0
1.6294 0 0.1321
0.1233 0 11.5821
MAP with negative correlation (MNC)
MAP with negative correlation (MNC)
MAP with negative correlation (MNC)
D0=
−55 0
0−50
00−40.5
,D1=
000
0.1504.85
40.30 0.2
MAP with zero correlation (MZC)
MAP with zero correlation (MZC)
MAP with zero correlation (MZC)
D0=
−10 1
0−10
00−5.25
,D1=
000
0.9500.05
0.15 0 5.1
The arrival process labeled
MPC
has correlated arrivals with the correlation between two successive
interarrival times given by 0.5315, the arrival process corresponding to the one labeled
MNC
has
correlated arrivals with the correlation between two successive interarrival times given by -0.4470 and
the arrival process labeled MZC has zero correlation between two successive interarrival times.
Service time follows continuous phase-type distribution (β
β
β,S)of order 6 in Model I.
Here we take β
β
β= (0.2,0.1,0.2,0.1,0.2,0.2)
S=
−6.70.501.200.5
0−5.50.50.210.8
00.1−5.50.90.31.2
0.100.8−6.50.30
0.300.50−4.51.3
0.20.300.40−5.5
.
In Model II, α′
α′
α′= (0.2,0.5,0.3),β′
β′
β′= (0.1,0.4,0.5),γ′
γ′
γ′=(α′
α′
α′,0
0
0)=(0.2,0.5,0.3,0,0,0).
S′=
−28.19 0.50
0−28.21 0.1
00.2−28.46
,T′=
−4.5583 0.30
0−4.582 0.3
00.1−4.6
,
L=T′T′0β′
β′
β′
0
0
0S′=
−4.5583 0.3 00.4258 1.7033 2.1292
0−4.582 0.30.4282 1.7128 2.1410
00.1−4.60.45 1.82.25
0 00−28.1900 0.50
0 00 0−28.2100 0.1
0 00 0 0.2−28..46
.
Service rate in Model I = Service rate in Model II= 3.7649. Fix
n
=6
,n
1
=3
,n
2
=3
,m
=3
,k
1
=
2,k
2=6,k =7,CK1 = 12,CK2 = 20, RS = 40.
Let ER1and ER2denote the expected revenue in Model I and Model II respectively.
https://doi.org/10.17993/3cemp.2022.110250.116-137
10.1 MAP with positive correlation (MPC) and the clock follows generalized Erlang distribution
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 51.2838 50.6032 8.0993 0.6468 7.4011 185.9164
13-13.5 48.9982 48.3254 7.8161 0.7322 7.5104 191.9783
14-14.5 46.7988 46.1337 7.5605 0.8178 7.5929 196.6326
15-15.5 44.6870 44.0296 7.3284 0.9032 7.6535 200.1340
16-16.5 42.6620 42.0121 7.1163 0.9881 7.6960 202.6849
17-17.5 40.7213 40.0788 6.9215 1.0722 7.7237 204.4475
18-18.5 38.8617 38.2264 6.7416 1.1555 7.7390 205.5523
19-19.5 37.0791 36.4510 6.5748 1.2378 7.7439 206.1053
20-20.5 35.3697 34.7486 6.4194 1.3190 7.7401 206.1931
206.1931
206.1931
21-21.5 33.7293 33.1151 6.2741 1.3990 7.7289 205.8866
22-22.5 32.1541 31.5466 6.1377 1.4778 7.7113 205.2444
23-23.5 30.6401 30.0393 6.0093 1.5554 7.6883 204.3152
24-24.5 29.1838 28.5895 5.8879 1.6316 7.6604 203.1393
25-25.5 27.7819 27.1941 5.7727 1.7065 7.6288 201.7505
26-26.5 26.4132 25.8498 5.6633 1.7800 7.5934 200.1771
Table 1 – Effect of θi’s on performance measures in Model I when the arrival process is MPC
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 19.3578 18.8077 6.0942 0.0214 7.0839 209.7992
14-14.5 15.2272 14.7401 5.6427 0.0350 7.7778 242.6997
16-16.5 11.8777 11.3801 5.2328 0.0524 8.3303 269.3707
18-18.5 9.168- 8.6946 4.8598 0.0374 8.7588 290.5665
20-20.5 7.0021 6.5515 4.5207 0.0977 9.0821 307.0833
22-22.5 5.3114 4.8821 4.2149 0.1248 9.3219 319.7984
24-24.5 4.0354 3.6254 3.9425 0.1546 9.5012 329.6455
26-26.5 3.1050 2.7124 3.7025 0.1868 9.6401 337.4388
28-28.5 2.4421 2.0649 3.4912 0.2211 9.7500 343.6853
30-30.5 1.9721 1.6087 3.3036 0.2573 9.8350 348.6118
32-32.5 1.6352 1.2844 3.1352 0.2951 9.8966 352.3382
34-34.5 1.3886 1.0493 2.9829 0.3342 9.9367 354.9907
36-36.5 1.2035 0.8749 2.8443 0.3742 9.9582 356.7130
37-37.5 1.1279 0.8044 2.7796 0.3946 9.9629 357.2698
38-38.5 1.0611 0.7425 2.7177 0.4151 9.9640 357.6454
39-39.5 1.0019 0.6879 2.6585 0.4357 9.9618 357.8543
40-40.5 0.9490 0.6395 2.6018 0.4565 9.9565 357.9098
357.9098
357.9098
41-41.5 0.9015 0.5964 2.5474 0.4774 9.9485 357.8242
42-42.5 0.8587 0.5578 2.4952 0.4983 9.9379 357.6082
43-43.5 0.8200 0.5231 2.4451 0.5194 9.9250 357.2720
44-44.5 0.7847 0.4918 2.3970 0.5405 9.9100 356.8247
45-45.5 0.7525 0.4634 2.3507 0.5616 9.8929 356.2746
Table 2 – Effect of θi’s on performance measures in Model II when the arrival process is MPC
Table 1 and Table 2 show the effect of
θ′
is
on various performance measures and the revenue function
in Model I and II respecively when the arrival process is MPC. In Model I, when
θ′
is
values increases,
the values of
ER1
increase and reach the maximum at
θ′
i
s=19-19.5 and then decreases. The maximum
revenue in this case is 206.1931. In Model II, when
θ′
is
values increases, the values of
ER2
increase and
reach the maximum at
θ′
is
= 40
−
40
.
5, and then decreases. The maximum value of
ER2
is 357.9098.
When we compare Model I and II, the values expected revenue in Model II is greater than that of the
corresponding values of expected revenue in Model I. Also, the values of the rate of perfect service
(RP) in Model II are greater than the corresponding values in Model I. In both models values of
RK1
decreases when
θ′
is
values increases. Also in both models,
RK2
increases when
θ′
is
values increases. This
is because when
θ′
is
values increase, the expected service time of the customer in each stage decreases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
131
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
10.2 MAP with positive correlation (MPC) and the clock follows Erlang distribution
Now we consider the case when all the values of θ′
isare equal.
θ′
isECS ECQ RK1RK2RP ER1
12 51.8634 51.1808 8.0856 0.6256 7.4584 188.7979
13 49.5574 48.8826 7.8021 0.7110 7.5679 194.8694
14 47.3366 46.6696 7.5466 0.7960 7.6500 199.5104
15 45.2033 44.5439 7.3147 0.8820 7.7100 202.9824
16 43.1571 42.5053 7.1028 0.9670 7.7517 205.4928
17 41.1959 40.5515 6.9084 1.0513 7.7784 207.2073
18 39.3166 38.6795 6.7289 1.1348 7.7926 208.2591
19 37.5154 36.8855 6.5626 1.2174 7.7964 208.7561
20 35.7882 35.1654 6.4076 1.2988 7.7914 208.7864
208.7864
208.7864
21 34.1312 33.5152 6.2628 1.3792 7.7790 208.4219
22 32.5401 31.9310 6.1270 1.4583 7.7603 207.7220
23 31.0113 30.4089 5.9990 1.5362 7.7362 206.7358
24 29.5411 28.9452 5.8781 1.6127 7.7074 205.5043
25 28.1260 27.5366 5.7635 1.6879 7.6746 204.0613
26 26.7629 26.1799 5.6545 1.7618 7.6381 202.4354
Table 3 – Effect of θi’s on performance measures in Model I when the arrival process is MPC
θ′
isECS ECQ RK1RK2RP ER2
12 19.4795 18.9287 6.1059 0.0197 7.0686 209.0797
14 17.2946 16.7574 5.8739 0.0257 7.4369 226.4730
16 15.3286 14.8048 5.6532 0.0327 7.7676 242.2119
18 9.2393 8.7652 4.8685 0.0700 8.7578 290.4893
20 7.0610 6.6097 4.5287 0.0937 9.0849 307.1775
22 5.3588 4.9288 4.2221 0.1204 9.3277 320.0353
24 4.0723 3.6617 3.9489 0.1498 9.5094 329.9940
26 3.1329 2.7397 3.7081 0.1816 9.6501 337.8737
28 2.4629 2.0852 3.4961 0.2156 9.7614 344.1917
30 1.9877 1.6238 3.3080 0.2515 9.8477 349.1815
32 1.6470 1.2957 3.1392 0.2891 9.9104 352.9645
34 1.3977 1.0580 2.9865 0.3280 9.9516 355.6660
36 1.2108 0.8818 2.8476 0.3678 9.9739 357.4294
37 1.1345 0.8105 2.7827 0.3881 9.9790 358.0040
38 1.0671 0.7479 2.7207 0.4085 9.9804 358.3955
39 1.0073 0.6928 2.6614 0.4291 9.9785 358.6188
40 0.9539 0.6440 2.6045 0.4499 9.9735 358.6872
358.6872
358.6872
41 0.9060 0.6005 2.5500 0.4707 9.9657 358.6131
42 0.8629 0.5615 2.4978 0.4917 9.9553 358.4074
43 0.8238 0.5265 2.4476 0.5127 9.9426 358.0803
44 0.7882 0.4949 2.3993 0.5338 9.9277 357.6409
45 0.7558 0.4663 2.3530 0.5549 9.9108 357.0978
Table 4 – Effect of θi’s on performance measures in Model II when the arrival process is MPC
Tables 3 and 4 show the effect of
θi
on performance measures and expected revenue (ER) when the
arrival process is MPC. In Model I ER is maximum at
θ
= 20 and the maximum revenue is 208.7864.
In Model II ER is maximum at
θ
= 40 and the maximum revenue is 358.6872. When
θi
’s values
increases, the values of
RK1
decrease at the same time the values of
RK2
increase. This is because the
expected service time of the customer in each stage decreases. When we compare Models I and II, the
values of expected revenue in Model II are greater than that of the corresponding values of expected
revenue in Model I. Also the values of the rate of perfect service (RP) in Model II are greater than the
corresponding values of RP in Model I.
10.3 MAP with negative correlation (MNC) and the clock follows generalized Erlang distribution
Tables 5 and 6 show the effect of
θ′
i
s on various performance measures and the revenue function when
the arrival process is MNC and the clock is a generalized Erlang clock.
ER
is maximum when
θi
’s
= 15
−
15
.
5in Model I and the maximum revenue is 273.7589. In Model II
ER
is maximum when
θi
’s = 40
−
40
.
5and the maximum revenue is 357.9432. When we compare Model I and II, the values
https://doi.org/10.17993/3cemp.2022.110250.116-137
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 18.1197 17.1597 11.4736 0.9058 10.4146 260.7828
13-13.5 12.6645 11.7209 11.0056 1.0202 10.5100 267.9301
14-14.5 9.2711 8.3467 10.5469 1.1295 10.5303 272.0594
15-15.5 7.0945 6.1913 10.1019 1.2330 10.4910 273.7589
273.7589
273.7589
16-16.5 5.6478 3.7669 9.6761 1.3309 10.4083 273.6003
17-17.5 4.6497 3.7914 9.2740 1.4239 10.2959 272.0712
18-18.5 3.9353 3.0993 8.8974 1.5120 10.1639 269.5493
19-19.5 3.4061 2.5918 .5466 1.5959 10.0197 266.3116
20-20.5 3.0018 2.2084 8.2205 1.6760 9.8681 262.5575
21-21.5 2.6844 1.9112 7.9172 1.7525 9.7122 258.4313
22-22.5 2.4294 1.6755 7.6350 1.8256 9.5543 254.0392
23-23.5 2.2204 1.4850 7.3718 1.8955 9.3958 249.4616
24-24.5 2.0461 1.3283 7.1260 1.9623 9.2380 244.7604
25-25.5 1.8985 1.1978 6.8959 2.0263 9.0815 239.9836
Table 5 – Effect of θi’s on performance measures in Model I when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 1.2636 0.6835 6.4269 0.0218 7.4496 220.4267
14-14.5 1.0887 0.5487 5.8233 0.0353 8.0099 249.8111
16-16.5 0.9632 0.3591 5.3212 0.0526 8.4581 273.4177
18-18.5 0.8684 0.2997 4.8976 0.0734 8.8175 292.4617
20-20.5 0.7938 0.2558 4.5356 0.0976 9.1056 307.8438
22-22.5 0.7335 0.2221 4.2230 0.1248 9.3354 320.2465
24-24.5 0.6834 0.1957 3.9503 0.1547 9.5174 330.1971
26-26.5 0.6410 0.1744 3.7105 0.1870 9.6594 338.1110
28-28.5 0.6046 0.1569 3.4980 0.2215 9.7681 344.3199
30-30.5 0.5728 0.1424 3.3083 0.2577 9.8486 349.0926
32-32.5 0.5449 0.1301 3.1381 0.2954 9.9054 352.6495
34-34.5 0.5200 0.1196 2.9845 0.3344 9.9418 355.1727
36-36.5 0.4977 0.1105 2.8451 0.3743 9.9611 356.8142
37-37.5 0.4874 0.1064 2.7802 0.3946 9.9650 357.3450
38-38.5 0.4775 0.1026 2.7182 0.4151 9.9655 357.7017
39-39.5 0.4681 0.0990 2.6588 0.4358 9.9630 357.8971
40-40.5 0.4592 0.0957 2.6020 0.4565 9.9574 357.9432
357.9432
357.9432
41-41.5 0.4506 0.0925 2.5476 0.4774 9.9492 357.8508
42-42.5 0.4424 0.0895 2.4954 0.4984 9.9386 357.6303
43-43.5 0.4346 0.0867 2.4452 0.5194 9.9256 357.2909
44-44.5 0.4270 0.0840 2.3971 0.5405 8.9104 356.8414
45-45.5 0.4197 0.0815 2.3508 0.5617 8.7875 356.2899
Table 6 – Effect of θi’s on performance measures in Model II when the arrival process is MNC
of expected revenue in Model II is greater than that of the corresponding values of expected revenue
in Model I. Also, the values of the rate of perfect service (RP) in Model II are greater than the
corresponding values in Model I. In both models values of
RK1
decreases when
θ′
is
values increases.
Also in both models,
RK2
increases when
θ′
is
values increases. This is because when
θ′
is
values increase,
the expected service time of the customer in each stage decreases.
10.4 MAP with negative correlation (MNC) and the clock follows Erlang distribution
Tables 7 and 8 show the effect of
θ
on various performance measures and expected revenue, when the
arrival process is MNC and the clock is an Erlang clock.
ER
is maximum at
θ
= 15 and the maximum
revenue is 278.5231 in Model I and
ER
is maximum at
θ
= 40 and the maximum revenue is 358.7241
in Model II.
10.5 MAP with zero correlation (MZC) and the clock follows generalized Erlang distribution
Tables 9 and 10 show the effect of
θ′
i
s on various performance measures and the revenue function
when the arrival process is MZC and the clock is generalized Erlang clock.
ER
is maximum when
θi
’s
= 16
−
16
.
5and the maximum revenue is 266.1353 in Model I and
ER
is maximum when
θi
’s = 40
−
40
.
5
and the maximum revenue is 350.9024 in Model II . When we compare Model I and II, the values
expected revenue in Model II is greater than the corresponding values of expected revenue in Model I.
Also, the values of the rate of perfect service (RP) in Model II are greater than the corresponding values
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
132
10.2 MAP with positive correlation (MPC) and the clock follows Erlang distribution
Now we consider the case when all the values of θ′
isare equal.
θ′
isECS ECQ RK1RK2RP ER1
12 51.8634 51.1808 8.0856 0.6256 7.4584 188.7979
13 49.5574 48.8826 7.8021 0.7110 7.5679 194.8694
14 47.3366 46.6696 7.5466 0.7960 7.6500 199.5104
15 45.2033 44.5439 7.3147 0.8820 7.7100 202.9824
16 43.1571 42.5053 7.1028 0.9670 7.7517 205.4928
17 41.1959 40.5515 6.9084 1.0513 7.7784 207.2073
18 39.3166 38.6795 6.7289 1.1348 7.7926 208.2591
19 37.5154 36.8855 6.5626 1.2174 7.7964 208.7561
20 35.7882 35.1654 6.4076 1.2988 7.7914 208.7864
208.7864
208.7864
21 34.1312 33.5152 6.2628 1.3792 7.7790 208.4219
22 32.5401 31.9310 6.1270 1.4583 7.7603 207.7220
23 31.0113 30.4089 5.9990 1.5362 7.7362 206.7358
24 29.5411 28.9452 5.8781 1.6127 7.7074 205.5043
25 28.1260 27.5366 5.7635 1.6879 7.6746 204.0613
26 26.7629 26.1799 5.6545 1.7618 7.6381 202.4354
Table 3 – Effect of θi’s on performance measures in Model I when the arrival process is MPC
θ′
isECS ECQ RK1RK2RP ER2
12 19.4795 18.9287 6.1059 0.0197 7.0686 209.0797
14 17.2946 16.7574 5.8739 0.0257 7.4369 226.4730
16 15.3286 14.8048 5.6532 0.0327 7.7676 242.2119
18 9.2393 8.7652 4.8685 0.0700 8.7578 290.4893
20 7.0610 6.6097 4.5287 0.0937 9.0849 307.1775
22 5.3588 4.9288 4.2221 0.1204 9.3277 320.0353
24 4.0723 3.6617 3.9489 0.1498 9.5094 329.9940
26 3.1329 2.7397 3.7081 0.1816 9.6501 337.8737
28 2.4629 2.0852 3.4961 0.2156 9.7614 344.1917
30 1.9877 1.6238 3.3080 0.2515 9.8477 349.1815
32 1.6470 1.2957 3.1392 0.2891 9.9104 352.9645
34 1.3977 1.0580 2.9865 0.3280 9.9516 355.6660
36 1.2108 0.8818 2.8476 0.3678 9.9739 357.4294
37 1.1345 0.8105 2.7827 0.3881 9.9790 358.0040
38 1.0671 0.7479 2.7207 0.4085 9.9804 358.3955
39 1.0073 0.6928 2.6614 0.4291 9.9785 358.6188
40 0.9539 0.6440 2.6045 0.4499 9.9735 358.6872
358.6872
358.6872
41 0.9060 0.6005 2.5500 0.4707 9.9657 358.6131
42 0.8629 0.5615 2.4978 0.4917 9.9553 358.4074
43 0.8238 0.5265 2.4476 0.5127 9.9426 358.0803
44 0.7882 0.4949 2.3993 0.5338 9.9277 357.6409
45 0.7558 0.4663 2.3530 0.5549 9.9108 357.0978
Table 4 – Effect of θi’s on performance measures in Model II when the arrival process is MPC
Tables 3 and 4 show the effect of
θi
on performance measures and expected revenue (ER) when the
arrival process is MPC. In Model I ER is maximum at
θ
= 20 and the maximum revenue is 208.7864.
In Model II ER is maximum at
θ
= 40 and the maximum revenue is 358.6872. When
θi
’s values
increases, the values of
RK1
decrease at the same time the values of
RK2
increase. This is because the
expected service time of the customer in each stage decreases. When we compare Models I and II, the
values of expected revenue in Model II are greater than that of the corresponding values of expected
revenue in Model I. Also the values of the rate of perfect service (RP) in Model II are greater than the
corresponding values of RP in Model I.
10.3 MAP with negative correlation (MNC) and the clock follows generalized Erlang distribution
Tables 5 and 6 show the effect of
θ′
i
s on various performance measures and the revenue function when
the arrival process is MNC and the clock is a generalized Erlang clock.
ER
is maximum when
θi
’s
= 15
−
15
.
5in Model I and the maximum revenue is 273.7589. In Model II
ER
is maximum when
θi
’s = 40
−
40
.
5and the maximum revenue is 357.9432. When we compare Model I and II, the values
https://doi.org/10.17993/3cemp.2022.110250.116-137
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 18.1197 17.1597 11.4736 0.9058 10.4146 260.7828
13-13.5 12.6645 11.7209 11.0056 1.0202 10.5100 267.9301
14-14.5 9.2711 8.3467 10.5469 1.1295 10.5303 272.0594
15-15.5 7.0945 6.1913 10.1019 1.2330 10.4910 273.7589
273.7589
273.7589
16-16.5 5.6478 3.7669 9.6761 1.3309 10.4083 273.6003
17-17.5 4.6497 3.7914 9.2740 1.4239 10.2959 272.0712
18-18.5 3.9353 3.0993 8.8974 1.5120 10.1639 269.5493
19-19.5 3.4061 2.5918 .5466 1.5959 10.0197 266.3116
20-20.5 3.0018 2.2084 8.2205 1.6760 9.8681 262.5575
21-21.5 2.6844 1.9112 7.9172 1.7525 9.7122 258.4313
22-22.5 2.4294 1.6755 7.6350 1.8256 9.5543 254.0392
23-23.5 2.2204 1.4850 7.3718 1.8955 9.3958 249.4616
24-24.5 2.0461 1.3283 7.1260 1.9623 9.2380 244.7604
25-25.5 1.8985 1.1978 6.8959 2.0263 9.0815 239.9836
Table 5 – Effect of θi’s on performance measures in Model I when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 1.2636 0.6835 6.4269 0.0218 7.4496 220.4267
14-14.5 1.0887 0.5487 5.8233 0.0353 8.0099 249.8111
16-16.5 0.9632 0.3591 5.3212 0.0526 8.4581 273.4177
18-18.5 0.8684 0.2997 4.8976 0.0734 8.8175 292.4617
20-20.5 0.7938 0.2558 4.5356 0.0976 9.1056 307.8438
22-22.5 0.7335 0.2221 4.2230 0.1248 9.3354 320.2465
24-24.5 0.6834 0.1957 3.9503 0.1547 9.5174 330.1971
26-26.5 0.6410 0.1744 3.7105 0.1870 9.6594 338.1110
28-28.5 0.6046 0.1569 3.4980 0.2215 9.7681 344.3199
30-30.5 0.5728 0.1424 3.3083 0.2577 9.8486 349.0926
32-32.5 0.5449 0.1301 3.1381 0.2954 9.9054 352.6495
34-34.5 0.5200 0.1196 2.9845 0.3344 9.9418 355.1727
36-36.5 0.4977 0.1105 2.8451 0.3743 9.9611 356.8142
37-37.5 0.4874 0.1064 2.7802 0.3946 9.9650 357.3450
38-38.5 0.4775 0.1026 2.7182 0.4151 9.9655 357.7017
39-39.5 0.4681 0.0990 2.6588 0.4358 9.9630 357.8971
40-40.5 0.4592 0.0957 2.6020 0.4565 9.9574 357.9432
357.9432
357.9432
41-41.5 0.4506 0.0925 2.5476 0.4774 9.9492 357.8508
42-42.5 0.4424 0.0895 2.4954 0.4984 9.9386 357.6303
43-43.5 0.4346 0.0867 2.4452 0.5194 9.9256 357.2909
44-44.5 0.4270 0.0840 2.3971 0.5405 8.9104 356.8414
45-45.5 0.4197 0.0815 2.3508 0.5617 8.7875 356.2899
Table 6 – Effect of θi’s on performance measures in Model II when the arrival process is MNC
of expected revenue in Model II is greater than that of the corresponding values of expected revenue
in Model I. Also, the values of the rate of perfect service (RP) in Model II are greater than the
corresponding values in Model I. In both models values of
RK1
decreases when
θ′
is
values increases.
Also in both models,
RK2
increases when
θ′
is
values increases. This is because when
θ′
is
values increase,
the expected service time of the customer in each stage decreases.
10.4 MAP with negative correlation (MNC) and the clock follows Erlang distribution
Tables 7 and 8 show the effect of
θ
on various performance measures and expected revenue, when the
arrival process is MNC and the clock is an Erlang clock.
ER
is maximum at
θ
= 15 and the maximum
revenue is 278.5231 in Model I and
ER
is maximum at
θ
= 40 and the maximum revenue is 358.7241
in Model II.
10.5 MAP with zero correlation (MZC) and the clock follows generalized Erlang distribution
Tables 9 and 10 show the effect of
θ′
i
s on various performance measures and the revenue function
when the arrival process is MZC and the clock is generalized Erlang clock.
ER
is maximum when
θi
’s
= 16
−
16
.
5and the maximum revenue is 266.1353 in Model I and
ER
is maximum when
θi
’s = 40
−
40
.
5
and the maximum revenue is 350.9024 in Model II . When we compare Model I and II, the values
expected revenue in Model II is greater than the corresponding values of expected revenue in Model I.
Also, the values of the rate of perfect service (RP) in Model II are greater than the corresponding values
https://doi.org/10.17993/3cemp.2022.110250.116-137
133
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
θ′
isECS ECQ RK1RK2RP ER1
12 19.9412 18.9776 11.4650 0.8768 10.5037 265.0314
13 13.7816 12.8336 11.0061 0.9923 10.6089 272.4372
14 9.9745 9.0450 10.5547 1.1029 10.6364 276.7431
15 7.5522 6.6436 10.1148 1.2078 10.6014 278.5231
278.5231
278.5231
16 5.9568 5.0703 9.6919 1.3071 10.5201 278.3574
17 4.8662 4.0022 9.2908 1.4012 10.4067 276.7548
18 4.0925 3.2510 8.9143 1.4905 10.2724 274.1164
19 3.5241 2.7044 8.5629 1.5754 10.1250 270.7381
20 3.0930 2.2945 8.2359 1.6565 9.9698 266.8317
21 2.7567 1.9786 7.9317 1.7338 9.8102 262.5493
22 2.4880 1.7294 7.6485 1.8077 9.6485 258.0019
23 2.2688 1.5288 7.3845 1.8784 9.4864 253.2722
24 2.0867 1.3646 7.1379 1.9460 9.3249 248.4235
25 1.9331 1.2282 6.9071 2.0106 9.1651 243.5047
Table 7 – Effect of θi’s on performance measures in Model I when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER2
12 1.2692 0.6879 6.4435 0.0200 7.4383 219.8071
14 1.0928 0.5517 5.8370 0.0330 8.0032 249.4221
16 0.9663 0.3609 5.3327 0.0497 8.4553 273.2238
18 0.8709 0.3010 4.9073 0.0700 8.8181 292.4344
20 0.7960 0.2567 4.5440 0.0936 9.1090 307.9595
22 0.7353 0.2229 4.2303 0.1204 9.3414 320.4850
24 0.6850 0.1963 3.9567 0.1499 9.5255 330.5411
26 0.6424 0.1749 3.7161 0.1818 9.6694 338.5454
28 0.6059 0.1574 3.5030 0.2159 9.7796 344.8316
30 0.5740 0.1428 3.3128 0.2519 9.8615 349.6701
32 0.5460 0.1305 3.1421 0.2893 9.9194 353.2827
34 0.5210 0.1199 2.9881 0.3281 9.9568 355.8528
36 0.4986 0.1108 2.8484 0.3679 9.9768 357.5335
37 0.4874 0.1067 2.7834 0.3882 9.9811 358.0814
38 0.4784 0.1029 2.7212 0.4086 9.9820 358.4535
39 0.4690 0.0993 2.6617 0.4292 9.9797 358.6628
40 0.4600 0.0959 2.6048 0.4499 9.9744 358.7241
358.7241
358.7241
41 0.4514 0.0927 2.5502 0.4707 9.9655 358.6404
42 0.4432 0.0897 2.4979 0.4917 9.9560 358.4299
43 0.4353 0.0869 2.4477 0.5127 9.9431 358.0995
44 0.4278 0.0842 2.3995 0.5338 9.9282 357.6579
45 0.4205 0.0817 2.3531 0.5549 9.9112 357.1132
Table 8 – Effect of θi’s on performance measures in Model II when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 16.8978 15.9692 11.963 0.8768 10.0764 252.3641
13-13.5 12.9656 12.0530 10.6422 0.9873 10.1672 259.2338
14-14.5 10.1494 9.2542 10.2102 1.0943 10.1985 263.5313
15-15.5 8.1129 7.2366 9.7986 1.1970 10.1805 265.6982
16-16.5 6.6205 5.7639 9.4070 1.2949 10.1229 266.1353
266.1353
266.1353
17-17.5 5.5094 4.6729 9.0355 1.3881 10.0346 265.1977
18-18.5 4.6677 3.8516 8.6844 1.4765 9.9233 263.1907
19-19.5 4.0190 3.2230 8.3536 1.5605 9.7956 260.3689
20-20.5 3.5102 2.7339 8.0430 1.6403 9.6565 256.9832
21-21.5 3.1044 2.3473 7.7517 1.7162 9.5102 253.0616
22-22.5 2.7757 2.0372 7.4789 1.7885 9.3595 248.8655
23-23.5 2.5057 1.7850 7.2233 1.8574 9.2069 244.4474
24-24.5 2.2807 1.5773 6.9837 1.9232 9.0538 239.8815
25-25.5 2.0912 1.4043 6.7591 1.9861 8.9014 235.2248
Table 9 – Effect of θi’s on performance measures in Model I when the arrival process is MZC
in Model I. In both models values of
RK1
decreases when
θ′
is
values increases. Also in both models,
RK2
increases when
θ′
is
values increases. This is because when
θ′
is
values increase, the expected service
time of the customer in each stage decreases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 1.2979 0.7292 6,3005 0.0213 7.3031 216.0909
14-14.5 1.0927 0.5633 5.7087 0.0346 7.8524 244.8974
16-16.5 0.9490 0.3537 5.2165 0.0516 8.2917 268.0396
18-18.5 0.8426 0.2848 4.8012 0.0720 8.6441 286.7090
20-20.5 0.7607 0.2350 4.4464 0.0956 8.9265 301.7886
22-22.5 0.6861 0.1942 4.0935 0.1115 9.0690 311.4098
24-24.5 0.6421 0.1691 3.8726 0.1517 9.3302 323.7022
26-26.5 0.5977 0.1466 3.6375 0.1834 9.4694 331.4603
28-28.5 0.5601 0.1285 3.4292 0.2171 9.5760 337.5471
30-30.5 0.5277 0.1138 3.2432 0.2526 9.6549 342.2260
32-32.5 0.4995 0.1016 3.0764 0.2896 9.7105 345.7129
34-34.5 0.4746 0.0913 2.9258 0.3278 9.7463 348.1865
36-36.5 0.4526 0.0827 2.7892 0.3670 9.7651 349.7957
37-37.5 0.4425 0.0788 2.7255 0.3869 9.7690 350.3161
38-38.5 0.4328 0.0752 2.6647 0.4070 9.7695 350.6657
39-39.5 0.4237 0.0719 2.6065 0.4272 9.7670 350.8573
40-40.5 0.4150 0.0688 2.5508 0.4475 9.7616 350.9024
350.9024
350.9024
41-41.5 0.4068 0.0659 2.4975 0.4680 9.7535 350.8119
42-42.5 0.3989 0.0632 2.4463 0.4886 9.7431 350.5957
43-43.5 0.3914 0.0607 2.3971 0.5092 9.7303 350.2630
44-44.5 0.3842 0.05583 2.3499 0.5299 9.7155 349.8224
45-45.5 0.3773 0.0561 2.3046 0.5506 9.6987 3492817.
Table 10 – Effect of θi’s on performance measures in Model II when the arrival process is MZC
θ′
isECS ECQ RK1RK2RP ER1
12 18.0958 17.1635 11.0909 0.8490 10.1654 256.5463
13 13.8212 12.9044 10.6417 0.9602 10.2618 263.5676
14 10.7643 9.8647 10.2140 1.0681 10.2973 267.9635
15 8.5599 7.6789 9.8059 1.1719 10.2821 270.1769
16 6.9502 6.0887 9.4169 1.2710 10.2259 270.6121
270.6121
270.6121
17 5.7566 4.9151 9.0473 1.3654 10.1376 269.6305
18 4.8564 4.0352 8.6972 1.4550 10.0253 267.5456
19 4.1654 3.3645 8.3669 1.5400 9.8956 264.6201
20 3.6258 2.8447 8.0563 1.6208 9.7540 261.0680
21 3.1973 2.4354 7.7647 1.6977 9.6047 257.0590
22 2.8514 2.1063 7.4914 1.7708 9.4510 252.7250
23 2.5682 1.8432 7.2352 1.8406 9.2950 248.1673
24 2.3331 1.6255 6.9951 1.9071 9.1387 243.4630
25 2.1355 1.4446 6.7698 1.9708 8.9831 238.6709
Table 11 – Effect of θi’s on performance measures in Model I when the arrival process is MZC
θ′
isECS ECQ RK1RK2RP ER2
12 1.3046 0.7347 6.3168 0.0197 7.2919 215.4835
14 1.0974 0.5670 5.7222 0.0324 7.8457 244.5159
16 0.9525 0.3558 5.2278 0.0487 8.2889 267.8495
18 0.8454 0.2863 4.8108 0.0686 8.6446 286.6823
20 0.7630 0.2361 4.4546 0.0918 8.9298 301.9020
22 0.6974 0.1987 4.1471 0.1180 9.1576 314.1811
24 0.6438 0.1699 3.8789 0.1469 9.3381 324.0394
26 0.5992 0.1472 3.6430 0.1782 9.4792 331.8862
28 0.5614 0.1290 3.4341 0.2117 9.5873 338.0488
30 0.5289 0.1142 3.2476 0.2469 9.6675 342.7921
32 0.5006 0.1020 3.0803 0.2837 9.7243 346.3337
34 0.4757 0.0917 2.9293 0.3217 9.7610 348.8537
36 0.4535 0.0829 2.7924 0.3607 9.7806 350.5009
37 0.4434 0.0791 2.7286 0.3805 9.7848 351.0379
38 0.4337 0.0755 2.6677 0.4006 9.7856 351.4027
39 0.4246 0.0721 2.6094 0.4207 9.7843 351.6079
40 0.4159 0.0690 2.5536 0.4411 9.7782 351.6654
351.6654
351.6654
41 0.4076 0.0661 2.5001 0.4615 9.7704 351.5859
42 0.3997 0.0643 2.4488 0.4820 9.7601 351.3796
43 0.3921 0.0609 2.3996 0.5026 9.7476 351.0556
44 0.3849 0.0585 2.3523 0.5233 9.7329 350.6227
45 0.3780 0.0563 2.3068 0.5440 9.7163 350.0888
Table 12 – Effect of θi’s on performance measures in Model II when the arrival process is MZC
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
134
θ′
isECS ECQ RK1RK2RP ER1
12 19.9412 18.9776 11.4650 0.8768 10.5037 265.0314
13 13.7816 12.8336 11.0061 0.9923 10.6089 272.4372
14 9.9745 9.0450 10.5547 1.1029 10.6364 276.7431
15 7.5522 6.6436 10.1148 1.2078 10.6014 278.5231
278.5231
278.5231
16 5.9568 5.0703 9.6919 1.3071 10.5201 278.3574
17 4.8662 4.0022 9.2908 1.4012 10.4067 276.7548
18 4.0925 3.2510 8.9143 1.4905 10.2724 274.1164
19 3.5241 2.7044 8.5629 1.5754 10.1250 270.7381
20 3.0930 2.2945 8.2359 1.6565 9.9698 266.8317
21 2.7567 1.9786 7.9317 1.7338 9.8102 262.5493
22 2.4880 1.7294 7.6485 1.8077 9.6485 258.0019
23 2.2688 1.5288 7.3845 1.8784 9.4864 253.2722
24 2.0867 1.3646 7.1379 1.9460 9.3249 248.4235
25 1.9331 1.2282 6.9071 2.0106 9.1651 243.5047
Table 7 – Effect of θi’s on performance measures in Model I when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER2
12 1.2692 0.6879 6.4435 0.0200 7.4383 219.8071
14 1.0928 0.5517 5.8370 0.0330 8.0032 249.4221
16 0.9663 0.3609 5.3327 0.0497 8.4553 273.2238
18 0.8709 0.3010 4.9073 0.0700 8.8181 292.4344
20 0.7960 0.2567 4.5440 0.0936 9.1090 307.9595
22 0.7353 0.2229 4.2303 0.1204 9.3414 320.4850
24 0.6850 0.1963 3.9567 0.1499 9.5255 330.5411
26 0.6424 0.1749 3.7161 0.1818 9.6694 338.5454
28 0.6059 0.1574 3.5030 0.2159 9.7796 344.8316
30 0.5740 0.1428 3.3128 0.2519 9.8615 349.6701
32 0.5460 0.1305 3.1421 0.2893 9.9194 353.2827
34 0.5210 0.1199 2.9881 0.3281 9.9568 355.8528
36 0.4986 0.1108 2.8484 0.3679 9.9768 357.5335
37 0.4874 0.1067 2.7834 0.3882 9.9811 358.0814
38 0.4784 0.1029 2.7212 0.4086 9.9820 358.4535
39 0.4690 0.0993 2.6617 0.4292 9.9797 358.6628
40 0.4600 0.0959 2.6048 0.4499 9.9744 358.7241
358.7241
358.7241
41 0.4514 0.0927 2.5502 0.4707 9.9655 358.6404
42 0.4432 0.0897 2.4979 0.4917 9.9560 358.4299
43 0.4353 0.0869 2.4477 0.5127 9.9431 358.0995
44 0.4278 0.0842 2.3995 0.5338 9.9282 357.6579
45 0.4205 0.0817 2.3531 0.5549 9.9112 357.1132
Table 8 – Effect of θi’s on performance measures in Model II when the arrival process is MNC
θ′
isECS ECQ RK1RK2RP ER1
12-12.5 16.8978 15.9692 11.963 0.8768 10.0764 252.3641
13-13.5 12.9656 12.0530 10.6422 0.9873 10.1672 259.2338
14-14.5 10.1494 9.2542 10.2102 1.0943 10.1985 263.5313
15-15.5 8.1129 7.2366 9.7986 1.1970 10.1805 265.6982
16-16.5 6.6205 5.7639 9.4070 1.2949 10.1229 266.1353
266.1353
266.1353
17-17.5 5.5094 4.6729 9.0355 1.3881 10.0346 265.1977
18-18.5 4.6677 3.8516 8.6844 1.4765 9.9233 263.1907
19-19.5 4.0190 3.2230 8.3536 1.5605 9.7956 260.3689
20-20.5 3.5102 2.7339 8.0430 1.6403 9.6565 256.9832
21-21.5 3.1044 2.3473 7.7517 1.7162 9.5102 253.0616
22-22.5 2.7757 2.0372 7.4789 1.7885 9.3595 248.8655
23-23.5 2.5057 1.7850 7.2233 1.8574 9.2069 244.4474
24-24.5 2.2807 1.5773 6.9837 1.9232 9.0538 239.8815
25-25.5 2.0912 1.4043 6.7591 1.9861 8.9014 235.2248
Table 9 – Effect of θi’s on performance measures in Model I when the arrival process is MZC
in Model I. In both models values of
RK1
decreases when
θ′
is
values increases. Also in both models,
RK2
increases when
θ′
is
values increases. This is because when
θ′
is
values increase, the expected service
time of the customer in each stage decreases.
https://doi.org/10.17993/3cemp.2022.110250.116-137
θ′
isECS ECQ RK1RK2RP ER2
12-12.5 1.2979 0.7292 6,3005 0.0213 7.3031 216.0909
14-14.5 1.0927 0.5633 5.7087 0.0346 7.8524 244.8974
16-16.5 0.9490 0.3537 5.2165 0.0516 8.2917 268.0396
18-18.5 0.8426 0.2848 4.8012 0.0720 8.6441 286.7090
20-20.5 0.7607 0.2350 4.4464 0.0956 8.9265 301.7886
22-22.5 0.6861 0.1942 4.0935 0.1115 9.0690 311.4098
24-24.5 0.6421 0.1691 3.8726 0.1517 9.3302 323.7022
26-26.5 0.5977 0.1466 3.6375 0.1834 9.4694 331.4603
28-28.5 0.5601 0.1285 3.4292 0.2171 9.5760 337.5471
30-30.5 0.5277 0.1138 3.2432 0.2526 9.6549 342.2260
32-32.5 0.4995 0.1016 3.0764 0.2896 9.7105 345.7129
34-34.5 0.4746 0.0913 2.9258 0.3278 9.7463 348.1865
36-36.5 0.4526 0.0827 2.7892 0.3670 9.7651 349.7957
37-37.5 0.4425 0.0788 2.7255 0.3869 9.7690 350.3161
38-38.5 0.4328 0.0752 2.6647 0.4070 9.7695 350.6657
39-39.5 0.4237 0.0719 2.6065 0.4272 9.7670 350.8573
40-40.5 0.4150 0.0688 2.5508 0.4475 9.7616 350.9024
350.9024
350.9024
41-41.5 0.4068 0.0659 2.4975 0.4680 9.7535 350.8119
42-42.5 0.3989 0.0632 2.4463 0.4886 9.7431 350.5957
43-43.5 0.3914 0.0607 2.3971 0.5092 9.7303 350.2630
44-44.5 0.3842 0.05583 2.3499 0.5299 9.7155 349.8224
45-45.5 0.3773 0.0561 2.3046 0.5506 9.6987 3492817.
Table 10 – Effect of θi’s on performance measures in Model II when the arrival process is MZC
θ′
isECS ECQ RK1RK2RP ER1
12 18.0958 17.1635 11.0909 0.8490 10.1654 256.5463
13 13.8212 12.9044 10.6417 0.9602 10.2618 263.5676
14 10.7643 9.8647 10.2140 1.0681 10.2973 267.9635
15 8.5599 7.6789 9.8059 1.1719 10.2821 270.1769
16 6.9502 6.0887 9.4169 1.2710 10.2259 270.6121
270.6121
270.6121
17 5.7566 4.9151 9.0473 1.3654 10.1376 269.6305
18 4.8564 4.0352 8.6972 1.4550 10.0253 267.5456
19 4.1654 3.3645 8.3669 1.5400 9.8956 264.6201
20 3.6258 2.8447 8.0563 1.6208 9.7540 261.0680
21 3.1973 2.4354 7.7647 1.6977 9.6047 257.0590
22 2.8514 2.1063 7.4914 1.7708 9.4510 252.7250
23 2.5682 1.8432 7.2352 1.8406 9.2950 248.1673
24 2.3331 1.6255 6.9951 1.9071 9.1387 243.4630
25 2.1355 1.4446 6.7698 1.9708 8.9831 238.6709
Table 11 – Effect of θi’s on performance measures in Model I when the arrival process is MZC
θ′
isECS ECQ RK1RK2RP ER2
12 1.3046 0.7347 6.3168 0.0197 7.2919 215.4835
14 1.0974 0.5670 5.7222 0.0324 7.8457 244.5159
16 0.9525 0.3558 5.2278 0.0487 8.2889 267.8495
18 0.8454 0.2863 4.8108 0.0686 8.6446 286.6823
20 0.7630 0.2361 4.4546 0.0918 8.9298 301.9020
22 0.6974 0.1987 4.1471 0.1180 9.1576 314.1811
24 0.6438 0.1699 3.8789 0.1469 9.3381 324.0394
26 0.5992 0.1472 3.6430 0.1782 9.4792 331.8862
28 0.5614 0.1290 3.4341 0.2117 9.5873 338.0488
30 0.5289 0.1142 3.2476 0.2469 9.6675 342.7921
32 0.5006 0.1020 3.0803 0.2837 9.7243 346.3337
34 0.4757 0.0917 2.9293 0.3217 9.7610 348.8537
36 0.4535 0.0829 2.7924 0.3607 9.7806 350.5009
37 0.4434 0.0791 2.7286 0.3805 9.7848 351.0379
38 0.4337 0.0755 2.6677 0.4006 9.7856 351.4027
39 0.4246 0.0721 2.6094 0.4207 9.7843 351.6079
40 0.4159 0.0690 2.5536 0.4411 9.7782 351.6654
351.6654
351.6654
41 0.4076 0.0661 2.5001 0.4615 9.7704 351.5859
42 0.3997 0.0643 2.4488 0.4820 9.7601 351.3796
43 0.3921 0.0609 2.3996 0.5026 9.7476 351.0556
44 0.3849 0.0585 2.3523 0.5233 9.7329 350.6227
45 0.3780 0.0563 2.3068 0.5440 9.7163 350.0888
Table 12 – Effect of θi’s on performance measures in Model II when the arrival process is MZC
https://doi.org/10.17993/3cemp.2022.110250.116-137
135
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
10.6 MAP with zero correlation (MZC)and the clock follows Erlang distribution
Tables 11 and 12 show the effect of
θ
on various performance measures and the revenue function, when
the arrival process is MZC and the clock, is an Erlang clock.
ER
is maximum at
θ
= 16 and the
maximum revenue is 270.6121 in Model I and
ER
is maximum when
θ
= 40 and the maximum revenue
is 351.6654 in Model II. Also, the values of the rate of perfect service (RP) in Model II are greater than
the corresponding values in Model I.
From Tables 1-12, we can conclude that in all cases, the values of
ER
and
RP
in Model II is greater
than the corresponding values of
ER
and
RP
in Model I. Moreover the values of
RK1
and
RK2
in
Model II are less than the corresponding values of RK1and RK2in Model I.
Figure 1 – Graph of Revenue Function
11 CONCLUSIONS
In this paper, we considered a
MAP/P H/
1queue. We analysed this model by using the matrix-analytic
method. We obtained the expected service time of a customer and also found the waiting time of
a tagged customer. Also, we constructed a revenue function and other performance measures. To
increase revenue, in Model II we consider the service time as the phase-type distribution (
γ′,L
)of
order
n
=
n1
+
n2
, which is the convolution of the two phase-type distributions (
α′,T′
)of order
n1
and (
β′,S′
)of order
n2
. We also performed some numerical experiments to evaluate some performance
measures and also found that the revenue is maximum in Model II.
ACKNOWLEDGMENT
The first author thanks Cochin University of Science and Technology for providing facilities to do her
Doctoral Programme.
https://doi.org/10.17993/3cemp.2022.110250.116-137
REFERENCES
[1]
Artalejo,J.R. (2000). G-networks: a versatile approach for work removal in queueing networks,
Europian Journel of Operation Research, 126, 233–249.
[2]
Bocharov,P.P.,Vishnevskii, V.M.(2003). G-Networks:development of the theory of multiplicative
networks,Automation and Remote Control, 64, 714–739.
[3]
Chakravarthy, S.R.(2009). A disaster queue with Markovian arrivals and impatient customers,
Applied Mathematics and Computation, 214, 48–59.
[4]
Gelenbe,E.(1991a). Product-form queueing networks with negative and positive customers, Journel
of Applied Probability, 28, 656–663.
[5]
Gelenbe,E.,Glynn,P.,Sigman,K. (1991b). Queues with negative arrivals, Journel of Appllied
Probability, 28, 245–250.
[6]
Klimenok, V., Dudin, A.N.(2012). A BMAP/PH/N Queue with Negative Customers and Partial
Protection of Service, Communications in Statistics—Simulation and Computation, 41, 1062–1082.
[7]
Krishnamoorthy, A. and Divya,V. (2018). (M,MAP)/(PH,PH)/1 Queue with Non-preemptive
Priority, Working Interruption and Protection, Reliability:theory and Applications, Vol.13, No.2(49).
[8]
Latouche,G. and Ramaswami,V. (1999). Introduction to Matrix Analytic Methods in Stochastic
Modelling, Philadelphia, ASA-SIAM.
[9]
Lucantoni, D.M.,Meier-Hellstern, K.S. and Neuts, M.F. (1990). A single-server queue with
server vacations and a class of nonrenewal arrival processes, Advances in Applied Probability, 22,
676-705.
[10]
Neuts, M.F. (1975). Computational Uses of The Method of Phases in the Theory of Queues,
Computer and Mathematics with Applications, Vol 1, Pergamon Press, Great Britian.
[11]
Neuts, M.F.(1979). A versatile Markovian point process. Journal of Applied Probability, 16,
764-779.
[12]
Neuts, M.F.(1981).Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,
The Johns Hopkins University Press, Baltimore.
[13]
Oliver, C.Ibe. (2009). Markov Processes for Stochastic Modeling, Elsevier Academic Press
Publications.
[14]
Qingqing Ye,Liwei liu (2018). Analysis of MAP/M/1 queue with working breakdowns, Com-
munications in Statistics-Theory and Methods, Vol.47, No13 3073-3084.
[15]
Sreenivasan,C.,Chakravarthy, S.R.,Krishnamoorthy A. (2013). MAP/PH/1 Queue with
working vacations, vacation interruptions and N-Policy,Applied Mathematical Modelling, Vol:37, No:6,
3879-3893.
https://doi.org/10.17993/3cemp.2022.110250.116-137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
136
10.6 MAP with zero correlation (MZC)and the clock follows Erlang distribution
Tables 11 and 12 show the effect of
θ
on various performance measures and the revenue function, when
the arrival process is MZC and the clock, is an Erlang clock.
ER
is maximum at
θ
= 16 and the
maximum revenue is 270.6121 in Model I and
ER
is maximum when
θ
= 40 and the maximum revenue
is 351.6654 in Model II. Also, the values of the rate of perfect service (RP) in Model II are greater than
the corresponding values in Model I.
From Tables 1-12, we can conclude that in all cases, the values of
ER
and
RP
in Model II is greater
than the corresponding values of
ER
and
RP
in Model I. Moreover the values of
RK1
and
RK2
in
Model II are less than the corresponding values of RK1and RK2in Model I.
Figure 1 – Graph of Revenue Function
11 CONCLUSIONS
In this paper, we considered a
MAP/P H/
1queue. We analysed this model by using the matrix-analytic
method. We obtained the expected service time of a customer and also found the waiting time of
a tagged customer. Also, we constructed a revenue function and other performance measures. To
increase revenue, in Model II we consider the service time as the phase-type distribution (
γ′,L
)of
order
n
=
n1
+
n2
, which is the convolution of the two phase-type distributions (
α′,T′
)of order
n1
and (
β′,S′
)of order
n2
. We also performed some numerical experiments to evaluate some performance
measures and also found that the revenue is maximum in Model II.
ACKNOWLEDGMENT
The first author thanks Cochin University of Science and Technology for providing facilities to do her
Doctoral Programme.
https://doi.org/10.17993/3cemp.2022.110250.116-137
REFERENCES
[1]
Artalejo,J.R. (2000). G-networks: a versatile approach for work removal in queueing networks,
Europian Journel of Operation Research, 126, 233–249.
[2]
Bocharov,P.P.,Vishnevskii, V.M.(2003). G-Networks:development of the theory of multiplicative
networks,Automation and Remote Control, 64, 714–739.
[3]
Chakravarthy, S.R.(2009). A disaster queue with Markovian arrivals and impatient customers,
Applied Mathematics and Computation, 214, 48–59.
[4]
Gelenbe,E.(1991a). Product-form queueing networks with negative and positive customers, Journel
of Applied Probability, 28, 656–663.
[5]
Gelenbe,E.,Glynn,P.,Sigman,K. (1991b). Queues with negative arrivals, Journel of Appllied
Probability, 28, 245–250.
[6]
Klimenok, V., Dudin, A.N.(2012). A BMAP/PH/N Queue with Negative Customers and Partial
Protection of Service, Communications in Statistics—Simulation and Computation, 41, 1062–1082.
[7]
Krishnamoorthy, A. and Divya,V. (2018). (M,MAP)/(PH,PH)/1 Queue with Non-preemptive
Priority, Working Interruption and Protection, Reliability:theory and Applications, Vol.13, No.2(49).
[8]
Latouche,G. and Ramaswami,V. (1999). Introduction to Matrix Analytic Methods in Stochastic
Modelling, Philadelphia, ASA-SIAM.
[9]
Lucantoni, D.M.,Meier-Hellstern, K.S. and Neuts, M.F. (1990). A single-server queue with
server vacations and a class of nonrenewal arrival processes, Advances in Applied Probability, 22,
676-705.
[10]
Neuts, M.F. (1975). Computational Uses of The Method of Phases in the Theory of Queues,
Computer and Mathematics with Applications, Vol 1, Pergamon Press, Great Britian.
[11]
Neuts, M.F.(1979). A versatile Markovian point process. Journal of Applied Probability, 16,
764-779.
[12]
Neuts, M.F.(1981).Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach,
The Johns Hopkins University Press, Baltimore.
[13]
Oliver, C.Ibe. (2009). Markov Processes for Stochastic Modeling, Elsevier Academic Press
Publications.
[14]
Qingqing Ye,Liwei liu (2018). Analysis of MAP/M/1 queue with working breakdowns, Com-
munications in Statistics-Theory and Methods, Vol.47, No13 3073-3084.
[15]
Sreenivasan,C.,Chakravarthy, S.R.,Krishnamoorthy A. (2013). MAP/PH/1 Queue with
working vacations, vacation interruptions and N-Policy,Applied Mathematical Modelling, Vol:37, No:6,
3879-3893.
https://doi.org/10.17993/3cemp.2022.110250.116-137
137
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
