ANALYSIS OF A DISCRETE TIME QUEUEING-INVENTORY
MODEL WITH BACK-ORDER OF ITEMS
M. P. Anilkumar
Department of Mathematics, T.M.Govt. College, Tirur-676502, Kerala, India
E-mail:anilkumarmp77@gmail.com
ORCID:0000-0002-7005-4843
K. P. Jose
Department of Mathematics, St.Peter’s College, Kolenchery - 682 311, Kerala, India.
E-mail:kpjspc@gmail.com
ORCID:0000-0002-8730-8539
Reception: 07/08/2022 Acceptance: 22/08/2022 Publication: 29/12/2022
Suggested citation:
M. P. Anilkumar and K. P. Jose (2022). Analysis is a discrete time queueing-inventory model with back-order of items.
3C Empresa. Investigación y pensamiento crítico,11 (2), 50-62. https://doi.org/10.17993/3cemp.2022.110250.50-62
https://doi.org/10.17993/3cemp.2022.110250.50-62
ABSTRACT
This paper analyses a discrete-time (
s, S
)queueing inventory model with service time and back-order
in inventory. The arrival of customers is assumed to be the Bernoulli process. Service time follows a
geometric distribution. As soon as the inventory level reaches a pre-assigned level due to demands, an
order for replenishment is placed. Replenishment time also follows a geometric distribution. When the
inventory level reduces to zero due to the service of customers or non-replenishment of items, a maximum
of
k
customers are allowed in the system and the remaining customers are assumed to be completely lost
till the replenishment. Matrix-Analytic Method (MAM) is used to analyze the model. Stability conditions,
various performance measures of the system, waiting-time distribution and reorder-time distribution are
obtained. Numerical experiments are also incorporated.
KEYWORDS
Discrete Time Queueing Inventory, Back-order, Cost Analysis, Matrix Analytic Method
https://doi.org/10.17993/3cemp.2022.110250.50-62
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
50
ANALYSIS OF A DISCRETE TIME QUEUEING-INVENTORY
MODEL WITH BACK-ORDER OF ITEMS
M. P. Anilkumar
Department of Mathematics, T.M.Govt. College, Tirur-676502, Kerala, India
E-mail:anilkumarmp77@gmail.com
ORCID:0000-0002-7005-4843
K. P. Jose
Department of Mathematics, St.Peter’s College, Kolenchery - 682 311, Kerala, India.
E-mail:kpjspc@gmail.com
ORCID:0000-0002-8730-8539
Reception: 07/08/2022 Acceptance: 22/08/2022 Publication: 29/12/2022
Suggested citation:
M. P. Anilkumar and K. P. Jose (2022). Analysis is a discrete time queueing-inventory model with back-order of items.
3C Empresa. Investigación y pensamiento crítico,11 (2), 50-62. https://doi.org/10.17993/3cemp.2022.110250.50-62
https://doi.org/10.17993/3cemp.2022.110250.50-62
ABSTRACT
This paper analyses a discrete-time (
s, S
)queueing inventory model with service time and back-order
in inventory. The arrival of customers is assumed to be the Bernoulli process. Service time follows a
geometric distribution. As soon as the inventory level reaches a pre-assigned level due to demands, an
order for replenishment is placed. Replenishment time also follows a geometric distribution. When the
inventory level reduces to zero due to the service of customers or non-replenishment of items, a maximum
of
k
customers are allowed in the system and the remaining customers are assumed to be completely lost
till the replenishment. Matrix-Analytic Method (MAM) is used to analyze the model. Stability conditions,
various performance measures of the system, waiting-time distribution and reorder-time distribution are
obtained. Numerical experiments are also incorporated.
KEYWORDS
Discrete Time Queueing Inventory, Back-order, Cost Analysis, Matrix Analytic Method
https://doi.org/10.17993/3cemp.2022.110250.50-62
51
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
1 INTRODUCTION
The first reported work on discrete-time queue was done by Meisling (1958). In that work, the researcher
analyzed a single-server queueing system in which interarrival time and service time are geometrically
distributed and as a limiting process, the results in a continuous system are derived. In queueing
inventory, stock out generates penalty costs due to the loss and disappointment of customers. This will
create perturbed demand and is considered in Schwartz (1966). Gross and Harris (1973) described the
progress of an (s, S) inventory system with complete back-ordering and state-dependent lead times.
The arrival is assumed to be a Poisson process and the lead time depends on the count of outstanding
orders. Archibald (1981) discussed the continuous review (s, S) Policies which minimize the average
stationary cost in an inventory system with constant lead time, fixed order cost, linear holding cost
per unit time, linear penalty cost per unit short, discrete compound Poisson demand, lost sales and
back-ordering. Krishnamoorthy and Islam (2004) analyzed an (s, S) Inventory model where demands
form a Poisson process. When the inventory level approaches zero due to service, upcoming arrivals are
transferred to a pool of finite capacity. Deepak et al. (2004) considered a queueing system in which
work gets postponed due to the finiteness of the buffer. When the buffer of finite capacity is full, further
demands are shifted to a pool of customers. A potential customer discovers the full buffer, and will opt
for the pool with some probability, or else it will be lost forever.
Manuel (2007) analysed a continuous perishable (s, S) inventory model in which the arrival is under
a Markovian arrival process. The expiry of items in the stock and the lead time follow independent
exponential distributions. Demands that arrive during a period when the items are out of stock enter
either a pool of finite capacity or are lost forever. Sivakumar (2007) studied a continuous perishable
inventory model in which the demand is following a Markovian arrival process. The inventoried items
have lifetimes that are assumed to follow an exponential distribution. The demands that occur during
stock-out periods either enter a pool that has a finite capacity or leaves the system. Any demand that
arrives when the pool is full and the inventory level is zero, is also assumed to be lost.
Sivakumar (2009) studied a continuous review perishable (s, S) inventory model in which the arrival
is in accordance with a Markovian arrival process. First In First Out discipline is used for the selection
of customers from the pool when the inventory level is above a pre-assigned positive value N, which is
at most the reorder level. The combined probability distribution of inventory level and the number of
demands, the system characteristics and expected total cost are obtained in the steady-state. Sivakumar
(2012) also considered a discrete-time inventory system in which demands follow a Markovian arrival
process. The replenishment of inventory is according to an (s, S) policy. The lead time follows a
phase-type distribution. The demands that take place in stock-out periods either enter a pool or go
away from the system with a pre-assigned probability. When the pool has no space and the inventory
level is dry, further demands that occur are considered to be lost. For a discussion of discrete-time
queueing models, one can refer to Alfa (2002, 2001) and Meisling (1958). The present paper generalizes
a work reported in the Ph.D. thesis of Deepthi (2013). The work in this paper is analysed by using the
Matrix-Analytic Method discussed in Neuts (1994).
The model in this paper has many applications in real-life situations. For instance, consider an
automobile showroom that accepts orders and delivers the vehicles whenever there are vehicles in stock.
Here the stock of vehicles can be considered inventory. If the items are exhausted due to service and
non-replenishment, orders of at most kare accepted and remaining demands are assumed to be lost.
The rest of the paper is organized as follows. Section 2 provides mathematical modeling and analysis.
The stability condition is derived in section 3. Steady-state probability vector and algorithmic analysis
are discussed in sections 4 and 5 respectively. Some relevant performance measures are included in
section 6. Section 7 analyses the waiting-time distribution of the potential customer. Reorder time
distribution is incorporated in section 8. Section 9 illustrates numerical experiments.
https://doi.org/10.17993/3cemp.2022.110250.50-62
2 Mathematical Modeling and Analysis
The following are the assumptions and notations used in this model.
Assumptions
(i) Inter-arrival times follow a geometric distribution with parameter p
(ii) Service time follows a geometric distribution with parameter q
(iii) Up to kcustomers are allowed in the system when the inventory level is zero
(iv) Lead time is geometrically distributed with parameter r
Notations
N(n):Number of customers in queue at an epoch n
I(n):Inventory level at the epoch n
Then
{
(
N
(
n
)
,I
(
n
));
n
=0
,
1
,
2
,
3
, ..}
is a Discrete Time Markov Chain(DTMC) with state space
{
(
i, j
);
i≥
0
,
0
<j≤S}∪{
(
i,
0) : 0
≤i≤k, }
. Now, the transition probability matrix of the process
has the form
P=
0123... k−1kk+1 k+2 ... .
0C1C00
1B2B1B0
20B2B1B0
3
.
.
..
.
..
.
..........
k−1B2B1B0
k .B
2D1D0
k+1 D2A1A0
.
.
.KA
2A1A0
.
.
.KA
2A1A0
.
.
.KA
2A1A0
.
.
..
.
..........
where the blocks C0,C
1,B
0,B
1,B
2,D
0,D
1,D
2, K, A0,A
1,and A2are given by
C0=
01...ss+1 ... S
0p¯r pr
1p¯r pr
.
.
....
sp¯r pr
s+1 p
.
.
....
Sp
C1=
01...ss+1 ... S
0¯p¯r¯pr
1¯p¯r¯pr
.
.
....
s¯p¯r¯pr
s+1 ¯p
.
.
....
S¯p
https://doi.org/10.17993/3cemp.2022.110250.50-62
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
52
1 INTRODUCTION
The first reported work on discrete-time queue was done by Meisling (1958). In that work, the researcher
analyzed a single-server queueing system in which interarrival time and service time are geometrically
distributed and as a limiting process, the results in a continuous system are derived. In queueing
inventory, stock out generates penalty costs due to the loss and disappointment of customers. This will
create perturbed demand and is considered in Schwartz (1966). Gross and Harris (1973) described the
progress of an (s, S) inventory system with complete back-ordering and state-dependent lead times.
The arrival is assumed to be a Poisson process and the lead time depends on the count of outstanding
orders. Archibald (1981) discussed the continuous review (s, S) Policies which minimize the average
stationary cost in an inventory system with constant lead time, fixed order cost, linear holding cost
per unit time, linear penalty cost per unit short, discrete compound Poisson demand, lost sales and
back-ordering. Krishnamoorthy and Islam (2004) analyzed an (s, S) Inventory model where demands
form a Poisson process. When the inventory level approaches zero due to service, upcoming arrivals are
transferred to a pool of finite capacity. Deepak et al. (2004) considered a queueing system in which
work gets postponed due to the finiteness of the buffer. When the buffer of finite capacity is full, further
demands are shifted to a pool of customers. A potential customer discovers the full buffer, and will opt
for the pool with some probability, or else it will be lost forever.
Manuel (2007) analysed a continuous perishable (s, S) inventory model in which the arrival is under
a Markovian arrival process. The expiry of items in the stock and the lead time follow independent
exponential distributions. Demands that arrive during a period when the items are out of stock enter
either a pool of finite capacity or are lost forever. Sivakumar (2007) studied a continuous perishable
inventory model in which the demand is following a Markovian arrival process. The inventoried items
have lifetimes that are assumed to follow an exponential distribution. The demands that occur during
stock-out periods either enter a pool that has a finite capacity or leaves the system. Any demand that
arrives when the pool is full and the inventory level is zero, is also assumed to be lost.
Sivakumar (2009) studied a continuous review perishable (s, S) inventory model in which the arrival
is in accordance with a Markovian arrival process. First In First Out discipline is used for the selection
of customers from the pool when the inventory level is above a pre-assigned positive value N, which is
at most the reorder level. The combined probability distribution of inventory level and the number of
demands, the system characteristics and expected total cost are obtained in the steady-state. Sivakumar
(2012) also considered a discrete-time inventory system in which demands follow a Markovian arrival
process. The replenishment of inventory is according to an (s, S) policy. The lead time follows a
phase-type distribution. The demands that take place in stock-out periods either enter a pool or go
away from the system with a pre-assigned probability. When the pool has no space and the inventory
level is dry, further demands that occur are considered to be lost. For a discussion of discrete-time
queueing models, one can refer to Alfa (2002, 2001) and Meisling (1958). The present paper generalizes
a work reported in the Ph.D. thesis of Deepthi (2013). The work in this paper is analysed by using the
Matrix-Analytic Method discussed in Neuts (1994).
The model in this paper has many applications in real-life situations. For instance, consider an
automobile showroom that accepts orders and delivers the vehicles whenever there are vehicles in stock.
Here the stock of vehicles can be considered inventory. If the items are exhausted due to service and
non-replenishment, orders of at most kare accepted and remaining demands are assumed to be lost.
The rest of the paper is organized as follows. Section 2 provides mathematical modeling and analysis.
The stability condition is derived in section 3. Steady-state probability vector and algorithmic analysis
are discussed in sections 4 and 5 respectively. Some relevant performance measures are included in
section 6. Section 7 analyses the waiting-time distribution of the potential customer. Reorder time
distribution is incorporated in section 8. Section 9 illustrates numerical experiments.
https://doi.org/10.17993/3cemp.2022.110250.50-62
2 Mathematical Modeling and Analysis
The following are the assumptions and notations used in this model.
Assumptions
(i) Inter-arrival times follow a geometric distribution with parameter p
(ii) Service time follows a geometric distribution with parameter q
(iii) Up to kcustomers are allowed in the system when the inventory level is zero
(iv) Lead time is geometrically distributed with parameter r
Notations
N(n):Number of customers in queue at an epoch n
I(n):Inventory level at the epoch n
Then
{
(
N
(
n
)
,I
(
n
));
n
=0
,
1
,
2
,
3
, ..}
is a Discrete Time Markov Chain(DTMC) with state space
{
(
i, j
);
i≥
0
,
0
<j≤S}∪{
(
i,
0) : 0
≤i≤k, }
. Now, the transition probability matrix of the process
has the form
P=
0123... k−1kk+1 k+2 ... .
0C1C00
1B2B1B0
20B2B1B0
3
.
.
..
.
..
.
..........
k−1B2B1B0
k .B
2D1D0
k+1 D2A1A0
.
.
.KA
2A1A0
.
.
.KA
2A1A0
.
.
.KA
2A1A0
.
.
..
.
..........
where the blocks C0,C
1,B
0,B
1,B
2,D
0,D
1,D
2, K, A0,A
1,and A2are given by
C0=
01...ss+1 ... S
0p¯r pr
1p¯r pr
.
.
....
sp¯r pr
s+1 p
.
.
....
Sp
C1=
01...ss+1 ... S
0¯p¯r¯pr
1¯p¯r¯pr
.
.
....
s¯p¯r¯pr
s+1 ¯p
.
.
....
S¯p
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
B0=
01... s s+1 ... S
0p¯r pr
1p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp
¯q
B1=
01... s s+1 ... S
0¯p¯r¯pr
1pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
....
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
....
S pq ¯p¯q
B2=
01... s s+1 ... S
00
1¯pq¯r¯pqr
.
.
....
s¯pq¯r¯pqr
s+1 ¯pq
.
.
....
S¯pq 0
D0=
12... s s+1 ... S
00 0
1p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp¯q
D1=
01... s s+1 ... S
0¯rr
1pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
....
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
....
S pq ¯p¯q
https://doi.org/10.17993/3cemp.2022.110250.50-62
D2=
01... s s+1 ... S
1q¯r qr
2¯pq¯r¯pqr
.
.
....
s+1 ¯pq¯r¯pqr
s+2 ¯pq
.
.
....
S¯pq 0
K=
01... s s+1 ... S
1q¯r qr
2
.
.
.
s
s+1
.
.
.
S
A0=
12... s s+1 ... S
1p¯q¯rp¯qr
2p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp¯q
A1=
12... s s+1 ... S
1¯p¯q¯r¯p¯qr
2pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
........
.
.
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
.......
S pq ¯p¯q
A2=
12... s s+1 ... S
10
2¯pq¯r¯pqr
.
.
.....
.
.
s¯pq¯r¯pqr
s+1 ¯pq
.
.
....
S¯pq 0
3 STABILITY AND STEADY-STATE ANAYSIS
Theorem 1. The above Markov chain is stable if and only if p<q
https://doi.org/10.17993/3cemp.2022.110250.50-62
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
54
B0=
01... s s+1 ... S
0p¯r pr
1p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp¯q
B1=
01... s s+1 ... S
0¯p¯r¯pr
1pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
....
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
....
S pq ¯p¯q
B2=
01... s s+1 ... S
00
1¯pq¯r¯pqr
.
.
....
s¯pq¯r¯pqr
s+1 ¯pq
.
.
....
S¯pq 0
D0=
12... s s+1 ... S
00 0
1p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp¯q
D1=
01... s s+1 ... S
0¯rr
1pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
....
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
....
S pq ¯p¯q
https://doi.org/10.17993/3cemp.2022.110250.50-62
D2=
01... s s+1 ... S
1q¯r qr
2¯pq¯r¯pqr
.
.
....
s+1 ¯pq¯r¯pqr
s+2 ¯pq
.
.
....
S¯pq 0
K=
01... s s+1 ... S
1q¯r qr
2
.
.
.
s
s+1
.
.
.
S
A0=
12... s s+1 ... S
1p¯q¯rp¯qr
2p¯q¯rp¯qr
.
.
....
sp¯q¯rp¯qr
s+1 p¯q
.
.
....
Sp
¯q
A1=
12... s s+1 ... S
1¯p¯q¯r¯p¯qr
2pq¯r¯p¯q¯r¯p¯qr +pqr
.
.
........
.
.
s pq¯r¯p¯q¯r¯p¯qr +pqr
s+1 pq ¯p¯q
.
.
.......
S pq ¯p¯q
A2=
12... s s+1 ... S
10
2¯pq¯r¯pqr
.
.
.....
.
.
s¯pq¯r¯pqr
s+1 ¯pq
.
.
....
S¯pq 0
3 STABILITY AND STEADY-STATE ANAYSIS
Theorem 1. The above Markov chain is stable if and only if p<q
https://doi.org/10.17993/3cemp.2022.110250.50-62
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Proof. Consider the matrix A=A0+A1+A2.Then
A=
1... s s+1 ... S
1q¯r¯q¯rr
.
.
.......
sq¯r¯q¯rr
s+1 q¯q
.
.
.......
Sq¯q
Let
π
be the steady state probability vector of A.Then the given Markov chain is stable if and only
if
π
(
A1
+2
A2
)
e>
1,where eis the column vector of ones of order
S
. On simplification, we get the
condition p<q.
STEADY STATE PROBABILITY VECTOR
Let x=(x0,x
1,...,x
k−1,x
k,...)be the steady state probability vector of P,
where xi=xi,j 0≤i≤k, 0≤j≤S
xi,j i>k,1≤j≤S
Under the stability condition, xi, for i≥k+1, is given by
xk+1+r=xk+1Rr,r≥0
where Ris the least non-negative root of the equation
R2A2+RA1+A0=R
with a value less than one for spectral radius. The vectors x0,x
1,...,x
k+1 are given by solving
x0C1+x1B2=x0
xJ−1B0+xjB1+xj+1B2=xj; (1 ≤j≤k−1)
xk−1B0+xKD1+xk+1[D2+RK(I−R)−1]=xk
xkD0+xk+1(A1+RA2)=xk+1
(1)
subject to the normalizing condition
k
i=0
xi+xk+1(I−R)−1e=1 (2)
EVALUATION OF THE TRUNCATION MATRIX R
The rate matrix
R
is given by
R
=
limn→∞ Rn
, where
Rn+1
=
�R2
nA2+A0
(
I−A1
)
−1and R0
=0.
The iteration is usually stopped when |(Rn+1 −Rn)|ij < ϵ, ∀i,j
COMPUTATION OF BOUNDARY PROBABILITIES
Now the system (1) can be solved using the block Gauss-Seidel iterative method. The vectors
x0,x
1,...,x
k+1sin the (n+ 1)th iteration are given by
x0(n+ 1) = x1(n)B2(I−C1)−1
xi(n+1)=[xi+1(n)B2+xi−1(n+ 1)B0](I−B1)−1; (1 ≤i≤k−1)
xk(n+1)=(xk−1(n+ 1)B0+xk+1(n)[D2+RK(I−R)−1])[I−D1]−1
xk+1(n+ 1) = xk(n+ 1)D0(I−A1−RA2)−1
Each iteration is subject to the normalizing condition (2).
https://doi.org/10.17993/3cemp.2022.110250.50-62
4 SYSTEM PERFORMANCE MEASURES
In order to consider some performance measures of the system under steady state, we take
xi,0
=0for
i>k
(i) Expected level of inventory, ELI, is given by
ELI =
S
j=1
∞
i=0
jxi,j
(ii) Expected number of customers, EC, is obtained by
EC =
S
j=1
∞
i=0
ixi,j
(iii) Expected departure after completing the service, EDS is given by
EDS =q
∞
i=1
∞
j=1
xi,j
(iv) The Expected reorder rate, ERR, is given by
ERR =q
∞
i=1
xi,s+1
(v) Expected replenishment rate
ERR =r
s
j=0
∞
i=0
xi,j
(vi) Probability that the inventory level zero, PI
0, is given by
PI
0=
∞
i=0
xi,0
(vii) Expected loss rate of customers
ELR =pxk0+q¯r
∞
i=k+1
(i−k)xi+1,1
(viii) Expected number of demands waiting in the system during stock out period, EW0is given by
EW0=
k
i=1
ixi,0
(ix) Expected reordering quantity, ERQ is given by
ERQ =
s
j=0
(S−j)yj,where yjis the probability that inventory level is j when
replenishment takes place
https://doi.org/10.17993/3cemp.2022.110250.50-62
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
56
Proof. Consider the matrix A=A0+A1+A2.Then
A=
1... s s+1 ... S
1q¯r¯q¯rr
.
.
.......
sq¯r¯q¯rr
s+1 q¯q
.
.
.......
Sq¯q
Let
π
be the steady state probability vector of A.Then the given Markov chain is stable if and only
if
π
(
A1
+2
A2
)
e>
1,where eis the column vector of ones of order
S
. On simplification, we get the
condition p<q.
STEADY STATE PROBABILITY VECTOR
Let x=(x0,x
1,...,x
k−1,x
k,...)be the steady state probability vector of P,
where xi=xi,j 0≤i≤k, 0≤j≤S
xi,j i>k,1≤j≤S
Under the stability condition, xi, for i≥k+1, is given by
xk+1+r=xk+1Rr,r≥0
where Ris the least non-negative root of the equation
R2A2+RA1+A0=R
with a value less than one for spectral radius. The vectors x0,x
1,...,x
k+1 are given by solving
x0C1+x1B2=x0
xJ−1B0+xjB1+xj+1B2=xj; (1 ≤j≤k−1)
xk−1B0+xKD1+xk+1[D2+RK(I−R)−1]=xk
xkD0+xk+1(A1+RA2)=xk+1
(1)
subject to the normalizing condition
k
i=0
xi+xk+1(I−R)−1e=1 (2)
EVALUATION OF THE TRUNCATION MATRIX R
The rate matrix
R
is given by
R
=
limn→∞ Rn
, where
Rn+1
=
�R2
nA2+A0
(
I−A1
)
−1and R0
=0.
The iteration is usually stopped when |(Rn+1 −Rn)|ij < ϵ, ∀i,j
COMPUTATION OF BOUNDARY PROBABILITIES
Now the system (1) can be solved using the block Gauss-Seidel iterative method. The vectors
x0,x
1,...,x
k+1sin the (n+ 1)th iteration are given by
x0(n+ 1) = x1(n)B2(I−C1)−1
xi(n+1)=[xi+1(n)B2+xi−1(n+ 1)B0](I−B1)−1; (1 ≤i≤k−1)
xk(n+1)=(xk−1(n+ 1)B0+xk+1(n)[D2+RK(I−R)−1])[I−D1]−1
xk+1(n+ 1) = xk(n+ 1)D0(I−A1−RA2)−1
Each iteration is subject to the normalizing condition (2).
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4 SYSTEM PERFORMANCE MEASURES
In order to consider some performance measures of the system under steady state, we take
xi,0
=0for
i>k
(i) Expected level of inventory, ELI, is given by
ELI =
S
j=1
∞
i=0
jxi,j
(ii) Expected number of customers, EC, is obtained by
EC =
S
j=1
∞
i=0
ixi,j
(iii) Expected departure after completing the service, EDS is given by
EDS =q
∞
i=1
∞
j=1
xi,j
(iv) The Expected reorder rate, ERR, is given by
ERR =q
∞
i=1
xi,s+1
(v) Expected replenishment rate
ERR =r
s
j=0
∞
i=0
xi,j
(vi) Probability that the inventory level zero, PI
0, is given by
PI
0=
∞
i=0
xi,0
(vii) Expected loss rate of customers
ELR =pxk0+q¯r
∞
i=k+1
(i−k)xi+1,1
(viii) Expected number of demands waiting in the system during stock out period, EW0is given by
EW0=
k
i=1
ixi,0
(ix) Expected reordering quantity, ERQ is given by
ERQ =
s
j=0
(S−j)yj,where yjis the probability that inventory level is j when
replenishment takes place
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4.1 WAITING TIME DISTRIBUTION
Here we assume the queue discipline as First In First Out (FIFO). Let
Vn
be the number of customers
in the queue ahead of the arriving customer. Then
((Vn,I(n))n=1,2,...)
is a discrete-time Markov
process with state space
{
(
i, j
)
,
0
≤i<∞,
0
≤j≤S}
, under the assumption that
k
is large. The transition probability matrix
is given by
P(Wq)=
10
tT
10
T2T1
......
......
,T1=
¯rr
¯q¯r¯qr
....
.
.
¯q¯r¯qr
¯q
...
¯q
(S+1)×(S+1)
,
T2=
00
¯q¯r¯qr
....
.
.
¯q¯r¯qr
¯q
...
¯q0
(S+1)×(S+1)
t=
0
q
.
.
.
q
(S+1)×1
Suppose there are
i
customers in the system in front of an arriving customer and
j
number of items
in the inventory is available to the arriving customer. Let T be the waiting time in queue. Then the
upper limit for T with a certain probability is calculated as follows.
Consider
zn
=
eij
[
P
(
Wq
)]
n
, where
eij
is infinite row vector whose (1 +
i
+
j
+
Si
)
th
element is one
remaining entries are zeros. Then P(T≤n)=zn(1) ,the first entry of zn
4.2 REORDER TIME DISTRIBUTION
In this section, we calculate the average time taken to fall inventory from
S
to
s
. If the current inventory
level is
s
+
i
, then re-order will takes place only when the service of
i
customers is completed. Note that
the distribution of time taken to reach inventory level to
s
from
S
is a discrete phase-type distribution
having (S−s)(S−s+ 3)/2phases and the transition probability matrix given by
P∗=10
tT
T=
s+1 s+2 ... S−N−1S−1S
s+1 H1
1
s+2 H2
2H2
1
.
.
.......
S−N−1
S−1HS−s−1
2HS−s−1
1
SH
S−s
2HS−s
1
where Hi
1=
01... i−1i
0¯pp
1¯p¯qp¯q
.
.
.......
i−1p¯qp¯q
iq
(i+1)×(i+1)
https://doi.org/10.17993/3cemp.2022.110250.50-62
Hi
2=
01... i−1
0 00
1¯pq pq
.
.
.......
i−1¯pq pq
iq
(i+1)×i
and t=
0
q1
.
.
.
0
[(S−s)(S−s+3)/2]×1
Therefore expected time to fall the inventory level from S to s due to service is
α
(
I−T
)
−11
, where 1is
the column vector of 1’s of order [(
S−s
)(
S−s
+ 3)
/
2]. If
α
is the [(
S−s
)(
S−s
+ 3)
/
2] row vector
whose
[((S−s)∗(S−s+ 1) + 2) /2]th
entry is 1 and remaining entries are zero, then expected time of
reorder is given by
ETR =α(I−T)−11
5 NUMERICAL EXPERIMENTS
5.1 COST FUNCTION
Define the expected total cost of the system per unit time as
ETC =C0ERO +C1(ERQ)(ERR)+C2(ELI)+C3(EW0)+C4(ELC)
where,
C0:The setup cost/order
C1:Procurement cost/unit
C2:Holding cost of inventory/unit/unit time
C3:Customers holding cost when inventory level is zero/customer/unit time
C4:Cost due to loss of customers/unit/unit time
5.2 GRAPHICAL ILLUSTRATIONS
In this paper, we obtained various performance measures. The change in the parameters such as arrival
rate, service rate, replenishment rate, number of back-order, etc. may affect these performance measures.
Figures 1, 2, and 3 illustrate the variation of
ETC
with
p
,
q
and
r
by keeping all other parameters
constant as indicated in the figure. The optimum value of
ETC
is obtained at
p
=0
.
5665 in figure 1,
q
=0
.
885 in figure 2 and
r
=0
.
1175 in figure 3, corresponding optimum values of the expected total
cost are 13.5276, 13.4706 and 14.6625 and which are indicated in the figures 1, 2 and 3 respectively.
In figure 4, we analyzed the variation of
ETW
using the expression for expected waiting time(
ETW
)
that we derived in the above section. By varying
q
and
r
, we analysed the variations of
ETW
, assuming
that arriving customers find 10 customers ahead of him and the level of inventory 2. We can see that
ETW decreases with the increase of qand r.
https://doi.org/10.17993/3cemp.2022.110250.50-62
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
58
4.1 WAITING TIME DISTRIBUTION
Here we assume the queue discipline as First In First Out (FIFO). Let
Vn
be the number of customers
in the queue ahead of the arriving customer. Then
((Vn,I(n))n=1,2,...)
is a discrete-time Markov
process with state space
{
(
i, j
)
,
0
≤i<∞,
0
≤j≤S}
, under the assumption that
k
is large. The transition probability matrix
is given by
P(Wq)=
10
tT
10
T2T1
......
......
,T1=
¯rr
¯q¯r¯qr
....
.
.
¯q¯r¯qr
¯q
...
¯q
(S+1)×(S+1)
,
T2=
00
¯q¯r¯qr
....
.
.
¯q¯r¯qr
¯q
...
¯q0
(S+1)×(S+1)
t=
0
q
.
.
.
q
(S+1)×1
Suppose there are
i
customers in the system in front of an arriving customer and
j
number of items
in the inventory is available to the arriving customer. Let T be the waiting time in queue. Then the
upper limit for T with a certain probability is calculated as follows.
Consider
zn
=
eij
[
P
(
Wq
)]
n
, where
eij
is infinite row vector whose (1 +
i
+
j
+
Si
)
th
element is one
remaining entries are zeros. Then P(T≤n)=zn(1) ,the first entry of zn
4.2 REORDER TIME DISTRIBUTION
In this section, we calculate the average time taken to fall inventory from
S
to
s
. If the current inventory
level is
s
+
i
, then re-order will takes place only when the service of
i
customers is completed. Note that
the distribution of time taken to reach inventory level to
s
from
S
is a discrete phase-type distribution
having (S−s)(S−s+ 3)/2phases and the transition probability matrix given by
P∗=10
tT
T=
s+1 s+2 ... S−N−1S−1S
s+1 H1
1
s+2 H2
2H2
1
.
.
.......
S−N−1
S−1HS−s−1
2HS−s−1
1
SH
S−s
2HS−s
1
where Hi
1=
01... i−1i
0¯pp
1¯p¯qp¯q
.
.
.......
i−1p¯qp¯q
iq
(i+1)×(i+1)
https://doi.org/10.17993/3cemp.2022.110250.50-62
Hi
2=
01... i−1
0 00
1¯pq pq
.
.
.......
i−1¯pq pq
iq
(i+1)×i
and t=
0
q1
.
.
.
0
[(S−s)(S−s+3)/2]×1
Therefore expected time to fall the inventory level from S to s due to service is
α
(
I−T
)
−11
, where 1is
the column vector of 1’s of order [(
S−s
)(
S−s
+ 3)
/
2]. If
α
is the [(
S−s
)(
S−s
+ 3)
/
2] row vector
whose
[((S−s)∗(S−s+ 1) + 2) /2]th
entry is 1 and remaining entries are zero, then expected time of
reorder is given by
ETR =α(I−T)−11
5 NUMERICAL EXPERIMENTS
5.1 COST FUNCTION
Define the expected total cost of the system per unit time as
ETC =C0ERO +C1(ERQ)(ERR)+C2(ELI)+C3(EW0)+C4(ELC)
where,
C0:The setup cost/order
C1:Procurement cost/unit
C2:Holding cost of inventory/unit/unit time
C3:Customers holding cost when inventory level is zero/customer/unit time
C4:Cost due to loss of customers/unit/unit time
5.2 GRAPHICAL ILLUSTRATIONS
In this paper, we obtained various performance measures. The change in the parameters such as arrival
rate, service rate, replenishment rate, number of back-order, etc. may affect these performance measures.
Figures 1, 2, and 3 illustrate the variation of
ETC
with
p
,
q
and
r
by keeping all other parameters
constant as indicated in the figure. The optimum value of
ETC
is obtained at
p
=0
.
5665 in figure 1,
q
=0
.
885 in figure 2 and
r
=0
.
1175 in figure 3, corresponding optimum values of the expected total
cost are 13.5276, 13.4706 and 14.6625 and which are indicated in the figures 1, 2 and 3 respectively.
In figure 4, we analyzed the variation of
ETW
using the expression for expected waiting time(
ETW
)
that we derived in the above section. By varying
q
and
r
, we analysed the variations of
ETW
, assuming
that arriving customers find 10 customers ahead of him and the level of inventory 2. We can see that
ETW decreases with the increase of qand r.
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3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
S= 20; s= 12; k= 6; ci=1for 0≤i≤4; q=0.6; r=0.1
p
0.48 0.5 0.52 0.54 0.56 0.58 0.6
ETC
13.525
13.53
13.535
13.54
13.545
13.55
13.555
13.56
13.565
13.57
p: 0.5665
ETC: 13.5276
Figure 1 – Variation of expected total cost with p
S= 20; s= 12; k= 6; ci=1,for 0≤i≤4; p=0.4; r=0.r
q
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93
ETC
13.47
13.475
13.48
13.485
13.49
13.495
13.5
q: 0.885
ETC: 13.4706
Figure 2 – Variation of expected total cost with q
https://doi.org/10.17993/3cemp.2022.110250.50-62
S= 20; s= 12; c0=c1=c2=1c3= 5; c4= 5; q=0.7; p=0.6;
r
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
ETC
14.5
15
15.5
16
16.5
17
r: 0.1175
ETC: 14.6625
Figure 3 – Variation of expected total cost with r
S= 10; s= 3; k= 12; p=0.4
q
0.45 0.5 0.55 0.6 0.65
ETW
15
16
17
18
19
20
21
22
23
24
25
r=0.2
r=0.3
r=0.5
Figure 4 – Variation of expected waiting time with q
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60
S= 20; s= 12; k= 6; ci=1for 0≤i≤4; q=0.6; r=0.1
p
0.48 0.5 0.52 0.54 0.56 0.58 0.6
ETC
13.525
13.53
13.535
13.54
13.545
13.55
13.555
13.56
13.565
13.57
p: 0.5665
ETC: 13.5276
Figure 1 – Variation of expected total cost with p
S= 20; s= 12; k= 6; ci=1,for 0≤i≤4; p=0.4; r=0.r
q
0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93
ETC
13.47
13.475
13.48
13.485
13.49
13.495
13.5
q: 0.885
ETC: 13.4706
Figure 2 – Variation of expected total cost with q
https://doi.org/10.17993/3cemp.2022.110250.50-62
S= 20; s= 12; c0=c1=c2=1c3= 5; c4= 5; q=0.7; p=0.6;
r
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
ETC
14.5
15
15.5
16
16.5
17
r: 0.1175
ETC: 14.6625
Figure 3 – Variation of expected total cost with r
S= 10; s= 3; k= 12; p=0.4
q
0.45 0.5 0.55 0.6 0.65
ETW
15
16
17
18
19
20
21
22
23
24
25
r=0.2
r=0.3
r=0.5
Figure 4 – Variation of expected waiting time with q
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CONCLUSIONS
In this paper, the attempt was to analyze a discrete-time inventory model with service time and
back-order in inventory. Stability condition, waiting time distribution and reorder time distribution are
analyzed. Numerical experiments are incorporated into the model to highlight the effect of variation
in system parameters. The work can be further extended by considering the Discrete Markov Arrival
Process(DMAP) and discrete phase type service distribution.
REFERENCES
[1] Alfa, A. S. (2002). Discrete time queues and matrix-analytic methods. Top, 10(2), 147-185.
[2]
Alfa, A. S. and Li, W. (2001). Matrix–geometric analysis of the discrete time Gi/G/1 system,
Stochastic Models, 541-554.
[3]
Archibald, B. C. (1981). Continuous review (s, S) policies with lost sales. Management Science,
27(10), 1171-1177.
[4]
Deepak, T. G.,Joshua, V. C. and Krishnamoorthy, A. (2004). Queues with postponed work.
Top, 12(2), 375-398.
[5]
Deepthi, C. P. and Krishnamoorthy, A. (2013). Discrete time inventory models with/without
positive service time Doctoral dissertation, Cochin University of Science and Technology.
[6]
Gross, D. and Harris, C. M. (1973). Continuous-Review (s, S) Inventory Models with State-
Dependent Leadtimes. Management Science, 19(5):567–574.
[7]
Krishnamoorthy, A. and Islam, M. E. (2004). (s, S) inventory system with postponed demands.
Stochastic analysis and applications, 22(3), 827-842.
[8]
Manuel, P.,Sivakumar, B. and Arivarignan, G. (2007). Perishable inventory system with
postponed demands and negative customers. Journal of Applied Mathematics and Decision Sciences,
2007.
[9] Meisling, T. (1958). Discrete-time queuing theory. Operations Research, 6(1), 96-105.
[10]
Neuts, M. F. (1994). Matrix-geometric solutions in stochastic models: an algorithmic approach.
Courier Corporation.
[11]
Schwartz, B. L. (1966). A new approach to stockout penalties. Management Science, 12(12),
B-538.
[12]
Manuel, P.,Sivakumar, B. and Arivarignan, G. (2007). Perishable inventory system with
postponed demands and negative customers. Journal of Applied Mathematics and Decision Sciences,
2007.
[13]
Sivakumar, B., Jayaraman, R. and Arivarignan, G. (2012). A discrete time inventory system
with postponed demands. Journal of Computational and Applied Mathematics, 236(13), 3073-3083.
https://doi.org/10.17993/3cemp.2022.110250.50-62
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