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ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2018, Vol. 57, No. 5, pp. 766–783. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © A.Z. Melikov, M.O. Shahmaliyev, 2018, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2018, No. 5.
SYSTEMS ANALYSIS
AND OPERATIONS RESEARCH
Markov Models of Inventory Management Systems
with a Positive Service Time
A. Z. Melikova,* and M. O. Shahmaliyeva
aInstitute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
*e-mail: agassi.melikov@gmail.com
Received May 7, 2017; in final form, May 7, 2018
Abstract—Markov models of perishable inventory systems where some customers are withdrawn after
completion of the service without buying the stock are studied. It is assumed that the service time and fill-
ing supply orders are positive random variables. The inventory replenishment size is a variable and depends
on the current inventory level. Exact and approximate methods for calculating the joint distribution of the
inventory level and the number of orders in the system are developed. Formulas for calculating the main
characteristics of the studied systems are proposed and the problem of their optimization is solved. The
high level of accuracy of the proposed formulas is confirmed by the numerical experiments.
DOI: 10.1134/S106423071805009X
INTRODUCTION
It is normally assumed that in the models of inventory systems (IS), the service time is zero (or negli-
gibly small) [1, 2]. At the same time, there are real systems where this assumption is not satisfied; i.e., it is
necessary to study IS models where the service time is a positive value. Such models were first studied in
[3, 4]. The conventional English term for them in the modern literature is queuing inventory systems
(QIS). The survey of publications devoted to QIS can be found in [5].
An important QIS subclass is perishable QIS (PQIS). Lately, PQIS models have been objects of
numerous studies. In the reviews [6–9] and in the monograph [10], there are references to many literary
sources on this subject.
Models of PQIS with a positive service time are studied in [11–16]. In [11], a model of the system with
a Poisson flow of customers is investigated, in which the order filling times and the inventory life time are
exponential distribution functions (d.fs.). There is a limited buffer (room) in the system for waiting in the
queue of customers; this system also receives a Poisson f low of negative customers, which do not require
inventory item after service completion. In order to obtain the system’s characteristics, a three-dimen-
sional Markov chain (3-D MC) is constructed. A similar model where the d.fs. of the service time for cus-
tomers are a phase-type distribution is studied in [12] using a 4-D MC; a similar model in the presence of
renewal orders is studied in [13] using 5-D MC. In order to study PQIS with a limited waiting buffer for
customers forming the Markov Arrival Process (MAP), a 6-D MC is constructed in [14] and a technique
for obtaining its characteristics is developed. A similar PQIS model for impatient customers is studied in
[15]. Here, methods of Markov decision-making processes are used to minimize a certain cost function
while assuming that the service rate is a controlled quantity. A similar problem is considered in [16].
It should be noted that in [11–16] it is assumed that inventory replenishment policies (IRPs) belong to
the class and they use a matrix geometric method to calculate the probability of states for multidi-
mensional Markov chains [17].
An important class of PQIS models includes those in which the server takes a break depending on the
current level of inventory and/or the number of orders in the system. Similar models are considered, for
example, in [18–20]. In these publications, 3-D MCs are used as mathematical models of PQIS with
server breaks and based on the method of phase state merge [21], hierarchical (approximate) algorithms
for the calculation of their state’s probabilities are proposed, and formulas are obtained for calculating the
characteristics of these systems.
As a rule, in QIS models it is assumed that after the completion of the service for each customer, the
system’s inventory level decreases. However, in practice, this assumption is not always correct. For QIS
models with durable inventories, this service scheme was first introduced in [22–24] independent of each
( ),s S
766
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 767
other. The authors give examples of real systems where the service of some orders does not lead to a reduc-
tion in the level of the system’s inventory. It is further applied to PQIS models in [25] and [26], while in
[25] it is assumed that the system uses the -policy (two-level policy), while in [26] an analogous
model with the -policy is studied.
At the same time, analyzing and optimizing PQIS models with more general inventory replenishment
policies are relevant problems today. One such policy is a policy where the size of the order depends on
the current level (variable size of the order (VSO)). This policy was first defined in [27, 28] where it was
applied to a QIS model with durable inventory, assuming that after the completion of servicing each cus-
tomer the system’s inventory level was reduced by a unit.
This article generalizes [25, 26], and here it is assumed that in a perishable inventory system a VSO pol-
icy of inventory replenishment with impatient customers is implemented. This generalization allows us to
perform a comparative analysis of the system’s characteristics using various IRPs and to choose the opti-
mal (in a given sense) policy. In this respect, the problems of conditional optimization of the studied mod-
els are solved here using different IRPs. To the best of our knowledge, PQIS models with different cus-
tomer types and VSO policies of inventory replenishment have not previously been investigated.
The mathematical models used for studying the considered systems are 2D Markov chains and in this
paper methods for their exact and approximate analysis are proposed. The approximate method allows us
to study models of almost any dimensionality, including models of infinite size. In this respect, it should
be noted that in order to address the issue of the “curse of dimensionality” in calculating the state proba-
bilities of high-dimensional 2-D MC researchers mostly employ the matrix analytic method [17] and the
spectral expansion solution [29, 30]. The use of these methods for the analysis of the studied models is
difficult for a number of reasons.
First, both methods impose certain structural requirements for the generating matrix of the studied
chain; in particular, for large numbers of rows (or columns) all elements of this matrix must be constant.
The satisfaction of this condition requires making some assumptions that are often inconsistent with the
actual condition of the system’s functioning. Thus, in the studied models, in order to fulfill the required
condition, it must be assumed that the dequeuing rate due to impatient customers does not depend on the
number of orders in the system, which is contrary to reality.
Secondly, both methods face problems of computational instability associated with the poor condi-
tionality of large-dimensional matrices used at various stages of these algorithms.
Third, these methods can only be applied for calculating the probabilities of 2-D MC states, which are
finite for at least one component. The method proposed by us is free of such shortcomings and in recent
years it has been successfully used to analyze various models of service systems [31, 32].
1. DESCRIPTION OF MODELS AND STATEMENT OF THE PROBLEM
In the studied system, which continuously monitors the inventory level of a limited size S. Each inven-
tory unit in it becomes unserviceable independently of the rest after a random time with an exponential
d.f. with parameter γ, > 0. At the same time, it is assumedthat the inventory item reserved while serving
the customer cannot perish. This means that the inventory level is reduced not only after the issue of the
item but also as a result of their perishing.
The system receives a Poisson flow of customers at the rate for the purchase of its inventory items.
For simplicity, we assume that incoming customers require a stock of unit size; i.e., after servicing such a
customer, the inventory level is reduced by a unit.
If the inventory level is positive at the time of the customer arrival, then it is accepted for service with
probability 1 provided that at that moment the server is free; otherwise, customers are queued. Customers
are entered in the queue according to the Bernoulli scheme even when the inventory level is zero; i.e., if
at the time of the next customer arrival in the system there are no stocks, then it is either queued with prob-
ability ϕ1, or with probability ϕ2 it leaves the system; at the same time, . In such cases, custom-
ers in the queue are impatient, i.e., every customer independently of the others waits in the queue for a
random period of time, which has an exponential distribution with the average of 
Here we consider systems with queues of finite and infinite lengths. In the model of systems with a
finite queue it is assumed that if at the time of the next customer arrival there are already N customers in
the system, it is dequeued with probability 1. In the model of systems with an infinite queue, all customers
can join the queue.
( ),s S
( )− 1,S S
γ
λ
ϕ + ϕ =1 2 1
−τ 1.
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
768 MELIKOV, SHAHMALIYEV
After the service of a customer has been completed, it, according to the Bernoulli scheme, either rejects
accepting the inventory item with probability or accepts the item with probability, 
Moreover, the service times of customers which refuse to accept the item have a general exponential d.f.
with the mean of ; the service times of customers which accept the item also have a general exponential
d.f. but with the mean of 
The inventory is replenished in accordance with the VSO policy, while the lead time is a positive ran-
dom variable that has an exponential distribution with the mean 
The variable size of the replenishment policy is determined as follows. A certain threshold value,
 is introduced and, if the inventory level exceeds this value, the system does not make orders for
inventory replenishment; and when the current level of the system’s inventory becomes equal to the value
s, it sends a replenishment order to the inventory of a higher level. At the same time, if at the moment of
the replenishment order delivery the current level of the system’s inventory is m, the size of the supply
depends on m and equals , 
We know the inventory replenishment policy of two levels (i.e., the -policy) is a particular case of
the proposed VSO policy. Indeed, if we assume that , and the value is determined by the
formula presented below, we obtain the policy of two levels, i.e.,
(1.1)
In order to be specific, as in the -policy, it is assumed that ; at the same time, the size of
the order size is obtained as
(1.2)
Ratio (1.2) means that when the inventory level drops to value m, an order is made of a size sufficient
for filling the system’s warehouse completely (the replenishment policy (1.2) was independently intro-
duced in the works [22, 27, 28]).
The problem consists in finding a joint distribution of the system’s inventory level and the number of
customers in the system. The solution of this problem allows us to determine the averaged performance
measures of the PQIS model under study and to perform its cost analysis.
2. METHODS FOR THE CALCULATION OF THE SYSTEM’S CHARACTERISTICS
The operation of a system with a finite queue is described by a two-dimensional Markov chain with
states of the form where m is the level of the warehouse inventory and n is the number of orders in
the system. The set of all possible states, i.e., the state space (SS) of this chain is determined as follows:
(2.1)
The graph of transitions between the states of this chain is shown in Fig. 1a.
The rate of transition from the state to another state is denoted by
. The set of these values determines the generating matrix (of the Q-matrix) of this 2-DMC.
Now we consider the problem of their calculation.
Transitions between the SS states (2.1) are associated with the following events: (i) customer arrival,
(ii) the completion of the customer service, (iii) expiration of the inventory’s lifetime (perishing),
(iv) dequeuing of the impatient customers, and (v) replenishment.
Taking into account the adopted inventory replenishment policy, the following three cases should be
distinguished in determining the initial state :
(1) m1 > s; (2) 0 < m1 ≤ s, (3) m1 = 0.
In cases where , transition from the state is impossible due to the events of type (iv) and
(v) since the customers in these states are patient and no replenishment takes place. The remaining tran-
sitions are determined as follows.
If an customer arrives (event of the type (i)) and the number of customers in the system is less than ,
then the number of orders in the system is increased by unit; i.e., there is a transition from the current state
σ1 σ2 σ + σ =1 2 1.
−μ 11
−μ μ < μ12, 2 1, .
−
v
1.
<, ,s s S
( )sd m < ≤ −0 ( ) .sd m S m
( , )s S
< /2s S ( )sd m
( ) − ≤⎧= ⎨ >⎩
, if ,
0, if .s
S s m s
d m
m s
( , )s S < /2s S
( ) − ≤⎧= ⎨ >⎩
, if ,
0, if .s
S m m s
d m
m s
( , )m n
= = ={( , ): 0, ; 0, }.E m n m S n N
∈1 1( , )m n E ∈2 2( , )m n E
1 1 2 2(( , ),( , ))q m n m n
∈1 1( , )m n E
>1m s 1 1( , )m n
N
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 769
Fig. 1. Graph of transitions between states of initial (a) and merged (b) model.
(S, N) (S, 2) (S, 1) (S, 0). . . .
(a)
(b)
(S, N − 1) (S, N − 2)
(S − 1, N)
(S − 1)γ Sγ
(S − 1, 2) (S − 1, 1) (S − 1, 0). . . .(S − 1, N − 1) (S − 1, N − 2)
(s, N) (s, 2) (s, 1) (s, 0). . . .(s, N − 1) (s, N − 2)
(1, N) (1, 2) (1, 1) (1, 0). . . .(1, N − 1) (1, N − 2)
(0, N) (0, 2)
Λ(1)Λ(s)Λ(S)
(0, 1) (0, 0). . . .
. . . . . .S
v
S − 1 s − 1s 1 0
(0, N − 1)
λϕ1
λ
τ
μ2σ2
Nτ
(0, N − 2)
μ1σ1
to the state E; the rate of this transition is . If after the order service has been completed, it
refuses to accept the goods (event of type (ii)), then the number of orders in the system is decreased by a
unit, while the system inventory level does not change; i.e., there is a transition from the current state to
the state E; the rate of this transition is . If after the service has been completed customer
purchase the item (event of type (ii)), then simultaneously the number of customers in the system and the
system’s inventory level are decreased by a unit; i.e., there is a transition from the current state to the state
 E; the rate of this transition is . After the item perishing (event of type (iii))
there is a transition from the current state to the state E; the rate of this transition is if
, and in the cases the rate of such a transition is .
From the above, we conclude that for the cases the elements of the Q-matrix are obtained as
follows (Fig. 1a):
(2.2)
If in the initial state the condition is satisfied, than the transition rates for the
above-mentioned events of types (i) and (iii) are obtained similarly to relations (2.2). In addition, at the
+ ∈1 1( , 1)m n λ
− ∈1 1( , 1)m n μ σ1 1
− − ∈1 1( 1, 1)m n ( )μ − σ2 11
− ∈1 1( 1, )m n γ1m
=1 0n >1 0n ( )− γ1 1m
>1m s
( )
λ = = +⎧
⎪μ σ = = −
⎪
μ σ = − = −⎪= ⎨ γ = − = =⎪
⎪ − γ = − > =
⎪
⎩
2 1 2 1
1 1 2 1 2 1
2 2 2 1 2 1
1 1 2 2
1 2 1 1 2
1 2 1 1 2 1
, if , 1,
, if , 1,
, if 1, 1,
(( , ),( , ))
, if 1, 0,
1 , if 1, 0, ,
0 otherwise.
m m n n
m m n n
m m n n
q m n m n
m m m n n
m m m n n n
∈1 1( , )m n E < ≤10 m s
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No.5 2018
770 MELIKOV, SHAHMALIYEV
arrival time of the replenishment order sized (event of type (v)), there is a transition from the state
 to the state ; and the rate of such a transition is .
Finally, let the condition be fulfilled in the initial state . In this case, transition from
this state due to events of types (ii) and (iii) is not possible because in these states customers cannot be
serviced and the system’s inventory is empty. In these states, the transition rates for events of types (i) and
(v) are obtained similarly to relations (2.2) and the arriving customer joins the queue with the probability
. The transition rates due to the customer’s impatience (event of type (iv)) are determined as follows:
after the impatient customer leaves the queue, the number of customers in the system is decreased by a
unit and moreover the system inventory level does not change; i.e., there is a transition from the current
state to the state E; the rate of this transition is Consequently, for the cases elements
of the Q-matrix are obtained as (see Fig. 1a)
(2.3)
Formulas (2.2) and (2.3) show that any state of this finite 2-D MC is reachable from any other state
(see Fig. 1a); i.e., there is a stationary mode in this system. Consequently, the probabilities of states
 are the unique solution of the system of equilibrium equations (SEEs) obtained based
on relations (2.2) and (2.3), which is also a system of linear algebraic equations with dimensionality
. Since the generation of this SEE is cumbersome and evident, its explicit form is not
given here.
The averaged performance measures of the studied PQIS are found from the state probabilities. We
have given below some of the frequently chosen main performance measures: the average inventory level
, the system’s average perishing rate , the average reorder rate , the probability of an cus-
tomer loss , and the average length of the queue . In using the proposed VSO policy, in addition
to the above-listed performance measures, a new performance measure must be introduced, i.e., the aver-
age size of replenishment order .
The average inventory level, the average size of a single order, and the average number of customers in
the system are determined as the mathematical expectations of the corresponding random variables; i.e.,
they are calculated from the following formulas:
(2.4)
(2.5)
(2.6)
Since the inventory item reserved while servicing the customer cannot perish the average perishing rate
is found as
(2.7)
Inventory is replenished whenever the system inventory level drops from the value down to the value s;
i.e., the average replenishment rate is calculated as
(2.8)
Customers are lost in the following cases: (1) if at the time of the arrival of the customer the queue
is full; (2) if the customer leaves the system due to its impatience in the absence of inventory item; (3)
if at the time of the arrival of the customer there are no inventory items in the system and the customer
− 1S m
1 1( , )m n 1( , )S n v
=1 0m ∈1 1( , )m n E
ϕ1
− ∈1(0, 1)n τ1 .n =1 0m
λϕ = = +⎧
⎪ τ = = −⎪= ⎨ν = =⎪
⎪⎩
1 2 2 1
1 2 2 1
1 2 2
2 2 1
, if 0, 1,
, if 0, 1,
((0, ),( , ))
, if , ,
0 otherwise.
m n n
n m n n
q n m n
m S n n
∈( , ), ( , )p m n m n E
( ) ( )= + +1 1E S N
( )avS Γ( )av ( )RR
( )PL ( )avL
( )avV
( )
= =
= ∑ ∑
1 0
, ;
S N
av
m n
S m p m n
( )
= − =
= −∑ ∑
0
, ;
S N
av
m S s n
V m p S m n
( )
= =
= ∑ ∑
1 0
, .
N S
av
n m
L n p m n
( ) ( ) ( )
= =
⎛ ⎛ ⎞⎞
Γ = γ + −⎜ ⎜ ⎟⎟
⎝ ⎝ ⎠⎠
∑ ∑
1 1
,0 1 , .
S N
av
m n
mp m m p m n
+ 1s
( ) ( ) ( ) ( )
=
= γ + + + μ σ + γ +∑2 2
1
1 1,0 1, .
N
n
RR s p s s p s n
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 771
refuses to wait in the queue. Consequently, we have the following formula for calculating the customer
loss probability:
(2.9)
It is impossible to find an analytical SEE solution due to the complex structure of its Q-matrix, and
the use of known numerical methods of linear algebra for this purpose is possible only for chains of a mod-
erate dimension, while they are generally useless for chains of large dimensionality.
This being so, this problem is solved here by the approximate method [25], which enables asymptotical
analysis of the performance measures of this PQIS model for a large-size inventory of the system and
queue buffer.
This method can be correctly used to study PQIS models, in which the rate of the customer arrival is
much higher than the perishing rates, i.e., It should be noted that this assumption is fully consistent
with the real operating condition of the PQIS since otherwise the system would not be able to fulfill its
purpose of serving the customers [33]. In addition, as it was stated above, 
If these assumptions are fulfilled, consider the following splitting of the SS (2.1):
(2.10)
where 
Splitting (2.10) means that the class of states contains those states from the original SS (2.1)
in which the inventory level is equal to m regardless of the number of customers in the system (i.e., the
state graph is partitioned by rows, see Fig. 1a). Based on the assumptions above, we conclude that the rates
of transitions between the states inside the rows far exceed the rates of transitions between the states from
different rows.
Further, based on splitting (2.10) in SS (2.1), the following merge function is determined:
(2.11)
where 〈m〉 denotes a merged state, which combines the class of states . Let us denote
Approximate values of the state probabilities of the initial model are obtained as
follows [25]:
(2.12)
where is the probability of a state inside the split model with the state space and is
the probability of the merged state .
Next we consider the problems of calculating the state probabilities of split and merged models. In all
states, inside the split model with SS Em the first component is constant and, therefore, all states of
this class are only determined by the second component. Thus, on studying split models with SS Em, each
state can only be given by the second component, i.e., as a matter of convenience on studying
the split model with SS Em its state is simply designated by 
From relations (2.2) we obtain that the probabilities of states inside all split models with SS 
coincide with the probabilities of states of the classical model with the load erl
(Erlang), i.e.,
(2.13)
From relations (2.3) we obtain that the probabilities of states inside the split model with SS coincide
with the probabilities of states of the classical Erlang model with the load erl, i.e.,
(2.14)
Here and below, the following notation is used: 
( ) ( ) ( )
− −
= = =
τ= + + ϕ
λϕ + τ∑ ∑ ∑
1 1
2
0 1 01
, 0, 0, .
S N N
m n n
nPL p m N p n p n
n
λ γ� .
μ μ�1 2.
=
= ∩ = ∅ ≠∪ 1 2 1 2
0
, , ,
S
m m m
m
E E E E m m
= ∈ = ={( , ) : 0, }, 0, .mE m n E n N m S
mE ( , )m n
= 〈 〉(( , )) ,U m n m
=, 0,mE m S
Ω = 〈 〉 ={ : 0, }.m m S
( ) ( ) ∈� , , ,p m n m n E
= ρ π 〈 〉�( , ) ( ) ( ),mp m n n m
ρ ( )m n ( , )m n ,mE π 〈 〉( )m
〈 〉 ∈ Ωm
( , )m n
∈( , ) mm n E
( , )m n =, 0, .n n N
=, 1,mE m S
/ /1/M M N = λ μ σ1 1a
( ) ( ) +ρ = − − =11 (1 ), 1, .n Nm n a a a m S
0E
/ / /0M M N = λϕ τ1b
( ) ( ) ( )
=
ρ = θ θ =∑0
0
, 0, .
N
j
n n j n N
( )θ = !.jj b j
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
772 MELIKOV, SHAHMALIYEV
The rate of transition from one merged state to another merged state is designated by
. According to [25], these parameters are obtained as follows:
(2.15)
With allowance for (2.2), (2.3), and (2.13)–(2.15), after certain transformations we find that these tran-
sition rates are calculated as (Fig. 1b):
(2.16)
where 
Then from (2.16) we obtain
(2.17)
where 
The probability is found from the normalization condition, i.e.,
(2.18)
Further with allowance for relations (2.13)–(2.18), we calculate the approximate values of the state
probabilities from (2.12) and taking them intoaccount we perform some transforma-
tions from (2.4)–(2.9), which yield formulas for an approximate calculation of the studied system’s per-
formance measures, i.e.,
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
Next, we consider a PQIS model with an infinite queue of customers. Its operation is also
described by 2-D MC but in this case, the SS of the model is infinite, i.e., the SS of this model is given as
(2.25)
Note 1. Here and below, for the sake of simplicity, the same notation is used in both models for their
SS, stationary distributions, and performance measures. Which particular model is discussed will be clear
from the context.
The elements of the Q-matrix of the given 2-D MC are obtained similarly to (2.2) and (2.3). The sys-
tem’s characteristics are calculated from relations (2.4)–(2.9); however, in these formulas it is necessary
〈 〉1m 〈 〉2m
〈 〉 〈 〉 〈 〉 〈 〉 ∈ Ω1 2 1 2( , ), ,q m m m m
( ) ( )( ) ( )
( )
( )
∈
∈
〈 〉 〈 〉 = ∑
1 1 1
2 2 2
1 2 1 1 2 2 1 1
,
,
( , ) , , , , .
m
m
m n E
m n E
q m m q m n m n p m n
( )Λ = −⎧
⎪
〈 〉 〈 〉 = ν ≤ =⎨
⎪⎩
1 2 1
1 2 1 2
, if 1,
( , ) , if , ,
0 otherwise,
m m m
q m m m s m S
( ) ( ) ( )( ) ( )( )Λ = γρ + − ρ μ σ + − γ =1 1 2 2 1 10 1 0 1 , 1, .m m m m S
( )
( )
ω π 〈 〉 ≤ ≤ +⎧π 〈 〉 = ⎨η π 〈 〉 + < ≤⎩
0 , 1 1,
( )
0 , 1 ,
m
m
m s
m
s m S
( )
( )
( )
( )
( )
( )
+
= =
ν + Λ − Λ + ν + Λ −ω = η =
Λ Λ Λ∏ ∏
1
1 1
1 1 1
; .
m s
m m
i i
i s i
i m i
( )π 〈 〉0
−+
= = +
⎛ ⎞
π 〈 〉 = + ω + η⎜ ⎟
⎝ ⎠
∑ ∑
11
1 2
( 0 ) 1 .
s S
m m
m m s
( ) ( ) ∈� , , ,p m n m n E
( )
=
≈ π 〈 〉∑
1
;
S
av
m
S m m
( ) ( ) ( ) ( )( )( )
=
Γ ≈ γ π 〈 〉 ρ + − − ρ∑
1
0 1 1 0 ;
S
av
m
m m m
( ) ( ) ( ) ( ) ( )( )( )≈ π 〈 + 〉 + γρ + γ + μ σ − ρ2 21 1 0 1 0 ;RR s s s
( ) ( )
=
= − π 〈 〉∑
0
;
s
av
m
V S m m
( )( ) ( ) ( ) ( ) ( ) ( )( )
−
=
⎛ ⎞τ≈ − π 〈 〉 ρ + π 〈 〉 ρ + ρ + ϕ − ρ⎜ ⎟λϕ + τ⎝ ⎠
∑
1
0 0 2 0
1 1
1 0 0 1 ;
N
n
nPL N N n N
n
( ) ( ) ( )( ) ( )
= =
≈ π 〈 〉 ρ + − π 〈 〉 ρ∑ ∑0
1 1
0 1 0 .
N N
av
n n
L n n n n
{ }= = =( , ) : 0, ; 0,1,... .E m n m S n
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 773
to take . Note that in this model, in calculating the order loss probability, the first term in (2.9) is
missing because the buffer for waiting orders is of an infinite size.
In this case, the stationary distribution of the corresponding infinite 2-D MC cannot be obtained using
the respective SEE for stationary state probabilities because here it is also impossible to find analytical
expressions for their calculation. Applying the method of two-dimensional generating functions is known
to involve certain difficulties. Therefore, this infinite 2-D MC has to be found by the above-described
technique. Here we also consider the splitting of SS (2.25) analogous to (2.10) and the merge function is
constructed accordingly (see (2.12)).
The probabilities of states inside all split models with SS , m > 0 are calculated as probabilities of
states of the model with the load erl, i.e.,
(2.26)
Note 2. It is assumed that the condition for the existence of a stationary mode in the system 
with the load erl, i.e., 
The probabilites of states inside the split model with SS are calculated as probabilities of states of
the model with the load erl, i.e.,
(2.27)
The state probabilities of the merged model are obtained similarly to (2.16)–(2.18). Further, using relations
(2.17), (2.18), (2.26), and (2.27), we find approximate values of the state probabilities of the given model.
After certain transformations, we obtain formulas for calculating the approximate values of the character-
istics of the studied PQIS model with an unlimited queue of customers. Thus, the average inventory level,
the system’s average perishing rate, and the average reorder rate are obtained by formulas (2.19)–(2.22),
respectively; at the same time, formulas (2.20) and (2.21) are further simplified because The
probability of customer loss and the average queue length in this model are calculated as follows:
(2.28)
(2.29)
Note 3. Formula (2.28) contains an infinite series and, unfortunately, it was not possible to find an explicit
formula for calculating its sum. At the same time, this series converges due to the convergence of the corre-
sponding majorant series . Therefore, here, for the approximate calculation of the sum
of this series, the upper limit of the sum is replaced by a sufficiently large value, which gradually increases and
this procedure continues until the value of the corresponding sum practically stops changing.
The above-described findings in the particular case (i.e., ) can yield results for the QIS
model with an infinite inventory lifetime.
3. NUMERICAL RESULTS
This section considers the results of numerical experiments carried out using the developed algorithms
for the approximate calculation of the performance measures of the high-dimension PQIS models. They
mainly study the behavior of these performance measures with respect to changes in the selected system
parameters in using different inventory replenishment policies.
Before presenting these results, let us consider the accuracy of the approximate values of the state prob-
abilities for the original two-dimensional Markov chain.
It is not possible to study the accuracy of the developed formulas analytically; thus, here we use the
comparative analysis of the results obtained by the numerical experiments. Here, the accuracy of the
approximate values of the state probabilities is estimated using the following norms (proximity measures):
maximum differences:
(3.1)
= ∞N
mE
∞/ /1/M M = λ μ σ1 1a
( ) ( )ρ = − =1 , 0,1,... .nn a a n
∞/ /1/M M
= λ μ σ1 1a < 1.a
0E
∞/ /M M = λϕ τ1b
( ) −ρ = =0 , 0,1,... .!
n
bbn e n
n
( )ρ = −0 1 .a
( )
∞
−
=
⎛ ⎞τ≈ π 〈 〉 + ϕ⎜ ⎟λϕ + τ⎝ ⎠
∑ 2
1 1
0 ;
!
n
b
n
b nPL e
n n
( ) ( )( )≈ π 〈 〉 + − π 〈 〉
−
0 1 0 .
1av
aL b
a
+ + +2 32 ! 3! ...b b b
−γ = ∞1 γ = 0
( ) ( )
∈
Ν = − �1 max ;E p pn n n
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
774 MELIKOV, SHAHMALIYEV
Table 1. Estimation of calculation accuracy for state probabilities with regard to various norms of proximity
Values of parameters Norms
s S N λ
1 10 10 20 0.019572 0.996411 0.859787
40 0.012387 0.999152 0.924913
60 0.008772 0.999626 0.948815
30 40 0.006512 0.998581 0.918092
60 0.00435 0.999611 0.951011
50 60 0.004831 0.99875 0.926706
70 60 0.008485 0.988533 0.820533
6 20 10 20 0.012668 0.995689 0.841564
40 0.00796 0.999029 0.915987
60 0.005631 0.999577 0.942885
30 40 0.004257 0.998506 0.911695
60 0.002826 0.999538 0.944291
50 40 0.005832 0.985151 0.80792
60 0.002826 0.998905 0.928744
70 60 0.00403 0.992309 0.859102
11 30 10 20 0.010171 0.995462 0.835242
40 0.00639 0.998987 0.91286
60 0.004521 0.99956 0.940802
30 40 0.003417 0.998503 0.909424
60 0.002268 0.999518 0.941928
50 40 0.004682 0.987063 0.825411
60 0.002268 0.998962 0.929457
70 60 0.003235 0.993323 0.87311
16 40 10 20 0.00875 0.995336 0.831666
40 0.005497 0.998962 0.911085
60 0.003889 0.99955 0.939618
30 40 0.00294 0.998504 0.908135
60 0.001951 0.999507 0.940586
50 40 0.004027 0.988068 0.835512
60 0.001951 0.998995 0.929864
70 60 0.002783 0.993855 0.881173
21 50 10 20 0.007799 0.995252 0.829281
40 0.004899 0.998946 0.9099
60 0.003466 0.999543 0.938827
30 40 0.00262 0.998505 0.907273
60 0.001739 0.9995 0.939689
50 40 0.00359 0.988716 0.842335
60 0.001739 0.999016 0.930135
70 60 0.00248 0.994197 0.886607
Ν 1 Ν 2 Ν 3
cosine similarity:
(3.2)( ) ( ) ( )( ) ( )( )
∈ ∈ ∈
⎛ ⎞ ⎛ ⎞Ν = ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∑ ∑ ∑� �
1/21/2
2 2
2 ;
E E E
p p p p
n n n
n n n n
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 775
Table 2. Estimation of calculation accuracy for characteristics (2.4), (2.7), and (2.8)
Values of parameters
s S N λ accurate approximate accurate approximate accurate approximate
1 10 [20,120][20,60] 2.447502 2.447502 3.589443 3.589443 0.512562 0.512562
6 20 [20,120] [20,60] 5.695907 5.695907 9.837346 9.837346 0.63643 0.63643
11 30 [20,120] [20,60] 8.826423 8.826423 16.010498 16.010498 0.672854 0.672854
16 40 [20,120] [20,60] 11.937077 11.937077 22.181819 22.181819 0.690602 0.690602
21 50 [20,120] [20,60] 15.040679 15.040679 28.355572 28.355572 0.701136 0.701136
avS Γav RR
Jaccard coefficient [34]:
(3.3)
Note 4. Since for the Euclidean norms
and the relation —where stands for the dimension of the set E—holds, they are
equivalent. Proceeding from this fact, it is proposed to use norm (3.1).
The results of a comparative analysis of the calculated state probabilities for the exact and approximate
approaches are shown in Table 1. Here, the initial data of the model are selected in the following way:
, and ϕ1 = 0.6.
The exact values of the state probabilities are calculated from the corresponding SEE using the well-
known algorithms of matrix decomposition whose complexity is estimated as . In other words, the
resources and time required for the SEE solution grow at a high rate with its increasing dimension. Thus,
for example, at S = 100 and N = 100 on a 4-core processor Corei7 2.40 GHz with 8GB RAM they will
require at least1 3–4 hrs. The experiments were carried out using the Apache Commons Math package
and the JAVA-based LU decomposition algorithm. It should be noted that by the approximate method the
performance measures of a model with the same dimension can be calculated on the same computer
in 3‒4 s.
From Table 1 it can be seen that the accuracy of the developed approximate formulas is quite high, with
the increase in the customer’s arrival rate, their accuracy increases with respect to all norms, i.e., with the
growing customer arrival rate, norm (3.1) approaches zero, while norms (3.2) and (3.3) approach unity. It
is clear from splitting (2.10) that with the increase in the arrival rate, the rates of transitions between the
state classes decrease; i.e., the smaller the transition rates between the state classes in the
assumed splitting the greater the accuracy of calculating the state probabilities of the initial model.
For the above-discussed initial data, we also analyzed the calculation accuracy for characteristics
(2.4)–(2.9) of the system (Tables 2, 3). It should be noted that the values of performance measures (2.4),
(2.7), and (2.8) obtained using the exact and approximate approaches almost completely coincide
(Table 2) and the small relative errors (less than 3%) are only observed in the calculation of performance
measures (2.9), which is acceptable in engineering calculations (Table 3). It should also be pointed out
that in a wide range of changes in the customer arrival rates, characteristics (2.4), (2.7), and (2.8) do not
depend on the size of the buffer of the queued orders.
The behavior displayed by the characteristics of the considered models with respect to the change in
the chosen system parameter is studied using approximate procedures since the developed approxi-
mate formulas are sufficiently accurate and their execution time is several seconds. To be more spe-
cific, here we study the behavior of these performance measures with respect to the change in the
critical level of inventory .
Since the policy is a particular case of the VSO policy (see formulas (1.2)), for brevity, we only pres-
ent the results of a comparative analysis of the system’s performance measures when these policies are used.
1 The text of the program can be downloaded from the following link: https://github.com/mamedshahmaliyev/SLE.
( ) ( ){ } ( ) ( ){ }
∈ ∈
Ν = ∑ ∑� �3 min , max , .
E E
p p p p
n n
n n n n
( ) ( )( )
∈
⎛ ⎞Ν = −⎜ ⎟
⎝ ⎠
∑ �
1
22
4
E
p p
n
n n
Ν ≤ Ν ≤ Ν1 4 1E E
μ = μ = γ = ν = τ = σ =1 2 115, 3, 2, 1, 0.5, 0.3
3( )O E
=, 0,1,..., ,mE m S
( )s
( ),s S
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
776 MELIKOV, SHAHMALIYEV
Table 3. Estimation of calculation accuracy for performance measures (2.6) and (2.9)
Values of parameters PL Lav
s S N λ accurate approximate accurate approximate
1 10 10 20 0.730948 0.795257 9.460112 9.605524
40 0.858358 0.89121 9.778314 9.831357
60 0.903661 0.925821 9.860933 9.893048
30 40 0.820976 0.855539 29.423218 29.450475
60 0.871208 0.893204 29.687109 29.7167
50 60 0.852899 0.877279 49.326578 49.309521
6 20 10 20 0.70913 0.787996 9.476948 9.642856
40 0.849987 0.88988 9.78693 9.846369
60 0.898765 0.925527 9.866474 9.902321
30 40 0.826006 0.866995 29.559085 29.602006
60 0.877944 0.904601 29.75495 29.789181
50 60 0.866198 0.894383 49.523637 49.527946
11 30 10 20 0.701413 0.785433 9.482975 9.656036
40 0.847031 0.889411 9.78998 9.851669
60 0.897036 0.925423 9.868433 9.905595
30 20 0.673959 0.774041 28.740021 28.516447
40 0.82778 0.87104 29.607078 29.655506
60 0.880322 0.908624 29.778907 29.814772
50 60 0.870893 0.900422 49.59322 49.605065
16 40 10 20 0.697023 0.783975 9.486405 9.663533
40 0.84535 0.889144 9.791715 9.854684
60 0.896053 0.925364 9.869547 9.907457
30 20 0.673397 0.774175 28.847506 28.683217
40 0.828789 0.87334 29.634376 29.685937
60 0.881675 0.910913 29.792533 29.829328
50 40 0.819089 0.86972 49.183994 48.969713
60 0.873563 0.903857 49.632799 49.64893
70 60 0.867787 0.901657 69.255649 69.066312
21 50 10 20 0.694085 0.782999 9.4887 9.668551
40 0.844224 0.888965 9.792877 9.856701
60 0.895395 0.925324 9.870293 9.908703
30 20 0.673024 0.774265 28.919349 28.794821
40 0.829464 0.87488 29.652644 29.706302
60 0.88258 0.912445 29.801653 29.839069
50 40 0.820819 0.871653 49.251232 49.067951
60 0.87535 0.906156 49.659285 49.678285
70 60 0.870202 0.904195 69.323142 69.159014
The model parameters are chosen as
It should be noted that the SS of this model contains 401 301 states, which makes it impossible to find
the exact values of the state probabilities from the respective SEE. Therefore, below we present the results
= = λ = μ = μ =
σ = ϕ = τ = γ = ν =
1 2
1 1
500, 800, 10, 15, 3,
0.3, 0.6, 0.5, 2, and 1.
S N
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 777
obtained using the developed approximate formulas (in these numerical experiments, the accuracy of the
approximate formulas is established by simulation modeling).
The results of the numerical experiments are shown in Fig. 2, where the notations 〈〈ο〉〉 and 〈〈×〉〉 on
the curves mark the results for the VSO policy and -policy, respectively. The presented analysis is
based solely on these data.
From Fig. 2a, it is clear that as the parameter increases, the function for both policies at first
grows and then for the -policy it slowly decreases and for the VSO policy it continues to grow at a
rather high rate. At the same time, at , the values of the function for both policies almost coin-
cide and for large values of the average inventory level under the VSO policy they appear greater than for
the -policy; at , the average inventory level of the VSO policy is greater by almost 30% than
( ),s S
s avS
( ),s S
≤ 50s avS
( ),s S > 240s
Fig. 2. Dependences of system’s performance measures on parameter in model with finite queue: (a), (b), (c),
 (d), (e), (f).
20
40
60
80
100
120
140
160
Sav
20 40 60 80 100 120 140 160
(a)
180 200 220 240
s
0
50
100
150
200
250
300
�av
20 40 60 80 100 120 140 160
(b)
180 200 220 240
s
0
100
200
300
400
500
Vav
20 40 60 80 100 120 140 160
(c)
180 200 220 240
s
0
s avS Γav avV
RR PL avL
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
778 MELIKOV, SHAHMALIYEV
Fig. 2. (Contd.)
0.2
0.4
0.6
0.8
RR
20 40 60 80 100 120 140 160
(d)
180 200 220 240
s
0
0.1
0.2
0.3
0.4
0.5
0.6
PL
20 40 60 80 100 120 140 160
(e)
180 200 220 240s0
100
200
300
400
500
600
700
800
Lav
20 40 60 80 100 120 140 160
(f)
180 200 220 240
s
0
for the -policy. The latter fact should be expected because when the VSO policy is used the size of
the suppliedinventory is always larger than when the -policy is pursued.
The behavior of the function corresponds to the behavior of the function (Fig. 2b), while the
intuitively assumed relation (see also formulas (2.4) and (2.7)) proves to be correct.
It should be noted that with an increase in the parameter , the accuracy of this solution is enhanced.
It can be seen from Fig. 2c that is an increasing function, while for values , the average
order size for the VSO policy is less than for the -policy, while for , the reverse picture is
observed. It should be noted that the graph of the function for the -policy is a segment that con-
nects points (0, 500) and (250, 250). This should be expected since for the -policy the order size is a
constant value; i.e., the relation holds (this graph again confirms the high level of accuracy of
the developed approximate formulas). At the same time, for the values , the average order size
( ),s S
( ),s S
Γav avS
( ) ( )Γ ≈ γav avs S s
s
avV ≤ ≤0 200s
( ),s S > 200s
avV ( ),s S
( ),s S
+ =avV s S
≤ ≤0 200s
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 779
for the VSO policy proves to be less than for the -policy, and at , the reverse situation is
observed.
The function RR is incremental with respect to the parameter for both policies (Fig. 2d). At the same
time, for the -policy, the inventory order frequency is higher than for the VSO policy. This is
explained by the fact that for the -policy, the size of the supplied inventory is smaller than for the
VSO policy.
The values of the function for different policies almost coincide (Fig. 2e). With an increase in the
parameter s, it decreases at a very low rate only for small values of the parameter s, i.e., at , and then
it becomes almost constant. Similar behavior is observed in the function , while the average queue
length is rather big and almost equal to 750 at the maximal size of the buffer 800 (Fig. 2f).
( ),s S > 200s
s
( ),s S
( ),s S
PL
≤ 20s
avL
Fig. 3. Dependences of system’s performance measures on parameter in model with infinite queue: (a), (b), (c),
 (d), (e), (f).
20
40
60
80
100
120
140
160
Sav
20 40 60 80 100 120 140 160
(a)
180 200 220 240
s
0
40
80
120
160
200
240
280
320
�av
20 40 60 80 100 120 140 160
(b)
180 200 220 240
s
0
40
80
160
320
120
200
240
280
Vav
20 40 60 80 100 120 140 160
(c)
180 200 220 240
s
0
s avS Γav avV
RR PL avL
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
780 MELIKOV, SHAHMALIYEV
Fig. 3. (Contd.)
0.2
0.4
0.6
0.8
0.7
0.5
0.3
0.1
RR
20 40 60 80 100 120 140 160
(d)
180 200 220 240
s
0
0.05
0.10
0.15
0.20
PL
20 40 60 80 100 120 140 160
(e)
180 200 220 240
s
0
1
2
3
4
5
Lav
20 40 60 80 100 120 140 160
(f)
180 200 220 240
s
0
The comparative analysis of the studied system’s characteristics for different inventory replenishment
policies shows that in terms of reducing the cost of the storage and inventory service, the use of the -
policy is effective; however, for the -policy, the order rate (frequency) is higher than for the VSO pol-
icy (each order is known to be associated with certain costs). In other words, in each specific case, it is
necessary to conduct a detailed analysis of various characteristics of the system and to determine the opti-
mal (in the relevant sense) inventory replenishment policy.
The results of the numerical experiments for the model with an infinite queue are shown in
Fig. 3, where the notations 〈〈×〉〉 and 〈〈ο〉〉 on the curves refer to the results obtained using the developed
approximate formulas and simulation modeling, respectively.
The initial data of the model are selected as before, except for the value ; i.e., it is assumed here that
 (this is done to satisfy the condition of the model’s ergodicity; i.e., it is required to satisfy the rela-
tion ). Obtaining reliable results (with a 5% confidence interval) using the simulation approach
( ),s S
( ),s S
( )= ∞N
λ
λ = 2.5
λ < μ σ1 1
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
MARKOV MODELS OF INVENTORY MANAGEMENT SYSTEMS 781
would require several tens of computer hours on devices with the above-mentioned characteristics, while
the execution time of the developed approximate algorithms is several seconds.
From the analysis of the graphs in Fig. 3, we conclude that the behavior of the system’s characteristics
obtained based on the approximate and simulation approaches are completely consistent with each other,
while their values are very close (sometimes they even coincide). The observed small errors are acceptable
in engineering calculations.
Interestingly, the comparative analysis of the results for a model with a finite (Fig. 2) and an infinite
queue (Fig. 3) shows that a quadruple decrease in the arrival rate and lifting the limit on the maximum
length of the queue leads to a significant improvement in performance measures (2.6) and (2.9), with the
other performance measures remaining almost unchanged. It should be noted that performance measures
(2.6) and (2.9) are related to the service system part, and the remaining performance measures are parts
of the inventory system.
The obtained formulas enable us to solve the optimization problems of this system. To be more spe-
cific, let us consider the task of minimizing the total fines (total cost, TC) associated with the functioning
of the system.
Suppose that the maximum sizes of the inventory and the queue buffer, as well as the load parameters
of the system, are fixed values and the only optimization parameter is the order point.
When the VSO inventory replenishment policy is used in the stationary mode, the total cost is obtained
as follows:
(3.4)
where is the fixed price of one order, is the unit price of the order size, is the unit inventory storage
price per unit of time, stands for the price of unit perishing, is the cost (penalty) for a single customer
loss, and is the price per unit time of queuing delay for a single customer.
Note 5. On using the -inventory replenishment policy, the total cost is determined according
to (3.4), where the multiplier is substituted for .
Note 6. Here and below in the optimization problem, the optimized parameter s is shown in brackets
in the expression of the functional and the left-hand side of limitations.
Let limitations be imposed on the average system inventory level and on the customer loss probability;
i.e., the following conditions should be fulfilled:
(3.5)
where and are the specified values. Then the optimization problem is reduced to finding such values
of the parameter that would provide the minimization of the total cost (3.4) when limitations (3.5) are
satisfied.
For any value of the input parameters, problem (3.4) and (3.5) has a solution since the set of possible
(admissible) solutions is discrete and finite. This problem can be solved by the standard method of finding
the minimum of a function from a discrete argument.
It should be noted that when using both inventory replenishment policies, the solution of problem (3.4)
and (3.5) for the model with a finite queue and the above-presented input data (if it exists) is . This
is explained by the fact that for the selected initial data all the components of functional (3.4) have the
minimum value at the point .
Table 4 shows the results of solving problem (3.4) and (3.5) for a model with an infinite queue with the
following initial data:
The coefficients in the expression for functional (3.4) were chosen from [33], i.e.,
In Table 4, the following notation is adopted: s* designates the optimal resolution of the problem (3.4)
and (3.5), is the minimum value of the functional (3.4), and the symbol means that problem (3.4)
and (3.5) does not have a solution.
An analysis of problem (3.4) and (3.5) shows that if the optimal solution of the problem exists for both
inventory replenishment policies,then these solutions are very close (sometimes they even coincide).
At the same time, for the VSO policy, the minimum value of functional (3.4) is much smaller than under
( ) ( )= + + + Γ + λ + ,r av h av ps av l w avTC s K c V RR c S c c PL c L
K rc hc
psc lc
wc
( ),s S
avV −S s
( ) ( )≤ ≤, ,avS s S PL s PL
S PL
s
= 0s
= 0s
= λ = μ = μ = σ = ϕ = τ = γ = ν =1 2 1 150, 1, 15, 3, 0.3, 0.6, 0.5, 0.2, and 1.S
= = = = = =10, 15, 0.3, 5, 3, 0.3.r h l w psK c c c c c
minTC ∅
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
782 MELIKOV, SHAHMALIYEV
Table 4. The results of solving problem (3.4), (3.5)
8 9 10 10 11 11 11 12 12 12 12 13 14
0.23 0.22 0.21 0.21 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.20 0.17
s* 0 1 2 ∅ 4 ∅ 3 21 24 6 ∅ ∅ ∅
VSO 0 ∅ 2 1 ∅ 3 ∅ ∅ ∅ 5 4 7 11
243 282 304 ∅ 332 ∅ 320 366 356 349 ∅ ∅ ∅
VSO 83 ∅ 133 112 ∅ 150 ∅ ∅ ∅ 180 166 205 249
S
PL
( , )s S
minTC ( , )s S
the -policy. In addition, the optimal solution of the problem, if it exists, is invariant in a sufficiently
large range of changes in the customer arrival rate. This fact is important for the practical implementation of
the optimal solution. The invariance of the optimal solution of problem (3.4) and (3.5) is explained by the
smooth change in the performance measures of system (2.4)–(2.9) with respect to the order arrival rate.
CONCLUSIONS
Markov models of perishable inventory management systems and the positive times of customer ser-
vice, fulfillment of orders, and the lifetime of a system’s inventory have been studied. Unlike in well-
known works, here it is assumed that some customers after the completion of their service fail to purchase
the system’s inventory item and in the absence of inventory items, the customers are queued according to
the Bernoulli scheme, while showing impatience in the queue. The inventory replenishment policy
belongs to the class of policies of a variable order size.
The accurate and approximate methods of determining the characteristics of the studied models with
a finite or infinite queue have been developed. The exact method is based on solving the balance equations
of two-dimensional Markov chains and it is effective for systems with a moderate size of the inventory and
the customer buffer. The approximate approach is based on phase merge algorithms for two-dimensional
Markov chains and can be applied to systems of any size.
Numerical experiments have been performed to compare the system’s performance measures for dif-
ferent inventory replenishment policies. A high level of accuracy of the developed approximate formulas
is shown. According to the results, the system’s performance measures that determine the components of
inventory management almost completely coincide for the accurate and approximate approaches, and
minor errors are only observed in calculating the performance measures which determine the components
of the service system.
Studying the system’s performance measures for various inventory replenishment policies shows that
the use of a policy of a variable size of restocking enables significantly improving some of these perfor-
mance measures compared to a policy with a fixed size of orders; however, this leads to a deterioration in
other performance measures. At the same time, the analyzed results of the problem on minimizing the sys-
tem’s total cost show that applying the policy of a variable order size makes it possible to substantially
reduce these costs.
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Translated by I. Pertsovskaya
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL Vol. 57 No. 5 2018
	INTRODUCTION
	1. DESCRIPTION OF MODELS AND STATEMENT OF THE PROBLEM
	2. METHODS FOR THE CALCULATION OF THE SYSTEM’S CHARACTERISTICS
	3. NUMERICAL RESULTS
	CONCLUSIONS
	REFERENCES