Keywords

1 Introduction

In many real life situations, customer, who needs inventoried items to complete his service, may arrive to service station according to Markovian arrival process. After that, he may go through different phases to complete his service in order to get the inventory. Moreover, common life time (CLT) for inventoried items may go through different phases until perishing, before they are taken by customers. The Markovian Arrival Process (MAP) is more general than Poisson process. MAP keeps the memoryless property of the Poisson process(partial memoryless) [3].

Many papers studied queueing inventory models. For example, Krishnamoorthy et al. [4] studied a PH/PH/1 queueing inventory system under (s, S) policy when the lead time is zero. AL Maqbali, Joshua and Krishnamoorthy [1] studied M/PH/1 queueing inventory system under (s, S) policy with lead time positive. Also, Krishnamoorthy and Shajin [5] studied the MAP/PH/1 queueing inventory system under (s, S) policy with lead time positive. In addition, Divya et al. [2] studied MAP/PH/1 queueing inventory system with processing of service items under vacation and N-policy with impatient customers. In their study, customers arrive according to MAP and service time follows two different phase type distribution. The inventory processing time follows phase type distribution. Moreover. Nair and Jose [8] studied the MAP/PH/1 production inventory model with varying service rates under (s, S) policy with lead time positive.

Some papers studied queueing inventory systems with common life time. For instance, Shajin et al. [10] studied a MAP/PH/1 queueing inventory system with Markovian lead time to bring the inventory level to its maximum. In their study, the common life time of inventoried items follows independent exponential distribution. Besides this, their study provided an interesting application of queueing inventory model with common life time as medicines with the same expiry date. Moreover, Shajin et al. [9] studied a MAP/M/1 and M/M/1 queueing inventory system with advanced reservation and cancellation for the next K time frames a head in the case of overbooking. In their study, the common life time (CLT) of inventoried items follows Phase type distribution.

Some papers studied queueing inventory systems with local purchase. Local purchase was introduced by Krishnamoorthy and Raju [6]. Krishnamoorthy, Varghese and Lakshmy [7] studied an (s, S) production inventory model with positive service time under local purchase.

As mentioned above, Krishnamoorthy and Shajin [5] and Divya et al. [2] studied MAP/PH/1 queueing inventory system. Then Shajin et al. [10] studied a MAP/PH/1 queueing inventory system with common life time. In this paper, we consider an MAP/PH/1 queueing inventory model under (s, S) policy with lead time positive. Besides this, we consider PH distributed common life time (CLT) for inventoried items. According to the common life time (CLT), the inventoried items perish all together. The supply of items is immediately in local purchase to bring the inventory level to the maximum inventory level S.

This model can be described as follows: customers arrive according to Markovian Arrival Process (MAP) with representation \((D_{0},D_{1})\) of order y. The service is assumed to follow PH-distribution with representation \((\beta ,T) \) of order m. An inventory of commodities is attached to the service station. The inventoried items have common life time (CLT) which follows PH-distribution with representation \((\alpha ,W) \) of order l. We assume that the inventoried items perish all together. In this model, the inventory is not allowed to go down to zero because of local purchase. In order to keep customer goodwill during stock out, the supply of items is immediately in local purchase to bring inventory level to S. Local purchases are purchased at a higher cost than the regular order (wait until replenishment) procedure.

Each service requires a unit of commodity for service. This unit is instantaneously taken at the beginning of the service. The replenishment of inventory follows (s, S) policy with lead time positive. The lead time follows exponential distribution with rate \(\theta \). When \( 1 \le i \le s\), the replenishment occurs to bring the inventory level i to S according to the rate of lead time. In the case of local purchase, the outstanding order of the normal purchase (wait until replenishment) is cancelled. According to (MAP), the first arriving customer instantaneously takes one item of the inventory at the beginning of his service. Then, the service of this customer immediately follows Phase type (PH) distribution. When service station is available, the next arriving customer takes one time at the beginning of his service and the service of this customer instantaneously follows Phase type (PH) distribution. Otherwise, this customer must wait in the buffer until the availability of service station. This process goes on.

According to types of blood group, blood bank has store for each blood group. The motivation for the model comes from the inventory management of one store in bank blood. For example, patients deal with one type of blood group in this store. They arrive according to Markovian Arrival Process (MAP). When the service station is available, the service of this patient follows Phase type (PH) distribution in the hospital and one blood bag is immediately taken from store to the patient at the beginning of his service. The common life time (CLT) for blood bags may go through different phases until perishing, before patients take the blood bags. In this case, the supply of blood bag is immediately in local purchase.

2 Mathematical Description of the Model

The model discussed above can be studied as a level Independent Qusi-Birth-Death (LIQBD) process. We introduce the following notations.

At time t:

  • N(t): the number of customers in the system.

  • I(t): the number of items in the inventory and these items are the same type.

  • L(t): the phase of common life time.

  • M(t): the phase of service.

  • Y(t): the phase of the arrival process.

\( X(t)=\{ (N(t), I(t),L(t),M(t),Y(t)); t\ge 0 \}\) is a continuous time Markov Chain (CTMC) with state space

\(\varOmega = \{ (0,i,l_{1},y_{1}); 1 \le i \le S; 1 \le l_{1} \le l; 1 \le y_{1} \le y \} \cup \{ (n,i,l_{1}, m_{1},y_{1}); n\ge 1; 1 \le i \le S; 1 \le l_{1} \le l; 1 \le m_{1} \le m ;1 \le y_{1} \le y \} \).

The terms of transitions of the states are shown in the Table 1.

Table 1. Intensities of transitions
Full size table

The infinitesimal generator Q of the continuous time Markov Chain (CTMC) is given by

$$ Q= \left( \begin{array}{ccccccccc} B_{00} &{}B_{01} &{} &{} &{} &{} &{} \\ B_{10}&{}A_{1} &{}A_{0}&{} &{} &{} &{} \\ &{}A_{2}&{}A_{1} &{} A_{0}&{} &{} &{} \\ &{} &{}A_{2} &{}A_{1} &{} A_{0} &{} &{} \\ &{} &{} &{} \ddots &{}\ddots &{}\ddots &{} \\ \end{array} \right) ; $$

where

\( B_{00}= \left( \begin{array}{ccc} \varUpsilon _{1} &{} O_{(yls) \times [(S-s-1)yl]} &{} \varUpsilon _{2} \\ O_{ ((S-1-s)yl) \times (syl)} &{} \varUpsilon _{3} &{} \varUpsilon _{4} \\ O_{ (yl) \times (syl)} &{} O_{ (yl) \times ((S-1-s)yl)} &{} \varUpsilon _{5} \end{array} \right) ; \)

\(B_{00}\) is a square matrix of order (Syl); 

where

$$\begin{aligned}&\varUpsilon _{1} = I_{(s)} \otimes ( (I_{l}\otimes D_{0}) + (W_{l} \otimes I_{l}) - [\theta I_{(yl)}]); \\&\varUpsilon _{2} = e_{s} \otimes [[(\alpha \otimes W_{l}^{0}) \otimes I_{y}] + \theta I_{ly} ] ; \\&\varUpsilon _{3}= I_{(S-1-s)} \otimes ( (I_{l}\otimes D_{0}) + (W_{l} \otimes I_{l}) ); \\&\varUpsilon _{4} = e_{(S-1-s)} \otimes [(\alpha \otimes W_{l}^{0}) \otimes I_{y}] \text { and }\\&\varUpsilon _{5} = (I_{l}\otimes D_{0}) + (W_{l} \otimes I_{l})+ [(\alpha \otimes W_{l}^{0}) \otimes I_{y}] . \end{aligned}$$
$$ B_{01} = \left( \begin{array}{ccc} O_{(yl) \times ((Syml)-(yml))} &{} I_{l} \otimes (\beta \otimes D_{1}) \\ I_{(S-1)} \otimes (I_{l} \otimes (\beta \otimes D_{1})) &{} O_{([S-1]yl) \times (ylm)} \end{array} \right) ; $$

\(B_{01}\) is a matrix of order \((Syl) \times (Syml).\)

$$ B_{10} = \left( \begin{array}{c} I_{S} \otimes [I_{l} \otimes (T^{0}_{m} \otimes I_{y})] \end{array} \right) ; $$

\( B_{10}\) is a matrix of order \( (Syml) \times (Syl)\).

$$A_{2}= \left( \begin{array}{cccc} O_{(yml) \times [(S-1)yml]} &{} [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ] \\ \varGamma &{} O_{[(S-1)yml] \times (yml)} \end{array} \right) ;$$

\( A_{2} \) is a square matrix of order (Syml);

where \( \varGamma = (I_{(S-1)} \otimes [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ])_{[(S-1)yml] \times [(S-1)yml]}. \)

$$ A_{0} = \left( \begin{array}{c} I_{(mlS)} \otimes D_{1}\\ \end{array} \right) ;$$

\( A_{0} \) is a square matrix of order (Syml).

$$ A_{1}= \left( \begin{array}{cc} \varphi _{1} &{} \varphi _{2}\\ (O_{[(S-1-s)yml] \times (ylms)} , \varphi _{3} ) &{} \varphi _{4} \\ O_{[yml] \times ([S-1]ylm)} &{} \varphi _{5} \end{array} \right) ;$$

\( A_{1} \) is a square matrix of order (Syml). where

\( \varphi _{1}= \left( \begin{array}{c } I_{s} \otimes ([ \{ I_{l} \otimes [(I_{m} \otimes D_{0})+(T_{m} \otimes I_{y})]\} + (W_{l} \otimes I_{(ym)})]-\theta I_{(yml)}) \end{array} \right) ;\)

\( \varphi _{1} \) is a matrix of order \( (syml) \times (ylms) \);

\( \varphi _{2}= \left( \begin{array}{c c } O_{(syml) \times [(S-s-1)ylm]}&(e_{s} \otimes [ [(\alpha \otimes W^{0}_{l})\otimes I_{(ym)}]+ \theta I_{(yml)} ])_{(syml) \times (yml)} \end{array} \right) ;\)

\( \varphi _{2}\) is a matrix of order \((syml) \times (yml)\);

\( \varphi _{3}= \left( \begin{array}{c } [I_{(S-1-s)} \otimes [\{ I_{l} \otimes [(I_{m} \otimes D_{0})+(T_{m} \otimes I_{y})]\} + (W_{l} \otimes I_{(ym)})]] \end{array} \right) .\)

\( \varphi _{3} \) is a matrix of order \(((S-1-s)yml) \times ((S-1-s)yml)\);

\( \varphi _{4}= \left( \begin{array}{c } [ e_{(S-1-s)} \otimes [(\alpha \otimes W^{0}_{l})\otimes I_{(ym)}]] \end{array} \right) ;\)

\( \varphi _{4} \) is a matrix of order \( ((S-1-s)yml) \times (yml) \);

\( \varphi _{5}= \left( \begin{array}{c } [\{ I_{l} \otimes [(I_{m} \otimes D_{0})+(T_{m} \otimes I_{y})]\} + (W_{l} \otimes I_{(ym)})] + [(\alpha \otimes W^{0}_{l})\otimes I_{(ym)}] \end{array} \right) ;\)

\( \varphi _{5} \) is a matrix of order \( (yml) \times (yml)\).

3 Steady-State Analysis

3.1 Stability Condition

Theorem 1

The stability condition of the queueing inventory model with common life time for inventoried items under study is given by

$$ \lambda < \mu $$

Where \(\lambda = (\sum _{i=0}^{(mlS)} \varvec{ \pi _{i}} ) D_{1} e_{y} \) ; \(\varvec{ \pi _{i} } \) is a row vector of order (y)

and \( \mu = (\sum _{i=0}^{S+1} {\varvec{ \pi _{i} }}) \wedge e_{(yml)} \) ;

where \( \wedge = [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ] \) and \( {\varvec{ \pi _{i} }} \) are row vectors of order (yml) .

Proof

Let \(A=A_{2} + A_{1} + A_{0}.\) We can realize that A is an irreducible matrix. Thus, there exists the stationary vector \(\varvec{\pi }\) of A such that

$$ {\varvec{ \pi }} A = 0 $$
$$ {\varvec{ \pi }} e = 1 .$$

The Markov chain with generator Q is stable if and only if

$$ {\varvec{ \pi }} A_{0} e < {\varvec{ \pi }} A_{2} e .$$

Recall, \( A_{0} = \left( \begin{array}{c} I_{(mlS)} \otimes D_{1}\\ \end{array} \right) ;\) \( A_{0} \) is a square matrix of order (Syml).

$$\begin{aligned} {\varvec{ \pi }} A_{0} e= & {} ({\varvec{ \pi _{0}}}, {\varvec{\pi _{1}}}, {\varvec{\pi _{2}}}, \dots , {\varvec{ \pi _{(mlS)}} }) \left( \begin{array}{ccccccccccc} D_{1} &{} &{} &{} \\ &{} D_{1} &{} &{} \\ &{} &{} \ddots &{} \\ &{} &{} &{} D_{1} \\ \end{array} \right) \left( \begin{array}{c} e_{y} \\ e_{y} \\ \vdots \\ e_{y}\\ \end{array} \right) _{(mlS)}; \\= & {} ( {\varvec{ \pi _{0}}} D_{1} ,{\varvec{\pi _{1}}} D_{1}, {\varvec{ \pi _{2}}} D_{1}, \dots , {\varvec{ \pi _{(mlS)}} } D_{1}) \left( \begin{array}{c} e_{y} \\ e_{y} \\ \vdots \\ e_{y}\\ \end{array} \right) _{(mlS)}; \\= & {} ({\varvec{ \pi _{0}}} D_{1} ,{\varvec{\pi _{1}}} D_{1}, {\varvec{ \pi _{2}}} D_{1}, \dots , {\varvec{ \pi _{(mlS)}} } D_{1}) \left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1\\ \end{array} \right) _{(mlS)} e_{y}; \\= & {} ({\varvec{ \pi _{0}}} D_{1} + {\varvec{\pi _{1}}} D_{1}+ {\varvec{ \pi _{2}}} D_{1}+ \dots + {\varvec{ \pi _{(mlS)}} } D_{1}) e_{y}; \\= & {} ({\varvec{ \pi _{0}}} + {\varvec{\pi _{1}}} + {\varvec{ \pi _{2}}} + \dots + {\varvec{(mlS)}} D_{1} e_{y}; \\= & {} (\sum _{i=0}^{(mlS)} {\varvec{\pi _{1}}}) D_{1} e_{y}\\= & {} \lambda . \end{aligned}$$

Recall,

$$A_{2}= \left( \begin{array}{cccc} O_{(yml) \times [(S-1)yml]} &{} [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ] \\ \varGamma &{} O_{[(S-1)yml] \times (yml)} \end{array} \right) ;$$

\( A_{2} \) is a square matrix of order (Syml).

where \( \varGamma = (I_{(S-1)} \otimes [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ])_{[(S-1)yml] \times [(S-1)yml]}. \)

To be more clear, we rewrite matrix \( A_{2} \) as following:

$$ A_{2} = \left( \begin{array}{ccccccccccccc} O &{} \cdots &{} &{} &{} &{} \wedge \\ \wedge &{} O &{} \cdots &{} &{} &{} O \\ O &{} \wedge &{} O &{} \dots &{} &{} \vdots \\ \vdots &{} O &{} \wedge &{}\\ \vdots &{} \vdots &{} &{}\ddots &{} &{} \vdots \\ &{} &{} &{} &{} \wedge &{} O\\ \end{array} \right) ;$$

where \( \wedge = [I_{l} \otimes [T^{0}_{m} \otimes (\beta \otimes I_{y})] ]. \)

$$\begin{aligned} {\varvec{ \pi }} A_{2} e= & {} ({\varvec{ \pi _{0}}}, {\varvec{\pi _{1}}}, {\varvec{ \pi _{2}}}, \dots , {\varvec{ \pi _{(S+1)}}}) \left( \begin{array}{ccccccccccccc} O &{} \cdots &{} &{} &{} &{} \wedge \\ \wedge &{} O &{} \cdots &{} &{} &{} O \\ O &{} \wedge &{} O &{} \dots &{} &{} \vdots \\ \vdots &{} O &{} \wedge &{}\\ \vdots &{} \vdots &{} &{}\ddots &{} &{} \vdots \\ &{} &{} &{} &{} \wedge &{} O\\ \end{array} \right) \left( \begin{array}{c} e_{yml} \\ e_{yml} \\ e_{yml}\\ \vdots \\ e_{yml}\\ e_{yml}\\ e_{yml}\\ \end{array} \right) _{(S)}; \\= & {} ( {\varvec{ \pi _{0} }} \wedge , {\varvec{\pi _{1}}} \wedge , {\varvec{ \pi _{2}}} \wedge , \dots , {\varvec{ \pi _{(S+1)}}} \wedge ) \left( \begin{array}{c} e_{yml} \\ e_{yml} \\ e_{yml}\\ \vdots \\ e_{yml}\\ e_{yml}\\ e_{yml}\\ \end{array} \right) _{(S)}; \\= & {} ( {\varvec{ \pi _{0}}} \wedge , {\varvec{\pi _{1}}} \wedge ,{\varvec{ \pi _{2}}} \wedge , \dots , {\varvec{ \pi _{(S+1)}}} \wedge ) \left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1\\ 1\\ \end{array} \right) _{(S)} e_{(yml)}; \\= & {} ( {\varvec{ \pi _{0}}} \wedge + {\varvec{\pi _{1}}} \wedge + {\varvec{ \pi _{2}}} \wedge + \dots + \varvec{ \pi _{(S+1)}} \wedge ) e_{(yml)}; \\= & {} ( {\varvec{ \pi _{0}}} + {\varvec{\pi _{1}}} + {\varvec{ \pi _{2}}} + \dots + {\varvec{ \pi _{(S+1)}}} ) \wedge e_{(yml)}; \\= & {} (\sum _{i=0}^{S+1} {\varvec{\pi _{1}}}) \wedge e_{(yml)}; \\= & {} \mu . \end{aligned}$$

Then, \( {\varvec{ \pi }} A_{0} e = (\sum _{i=0}^{(mlS)} {\varvec{\pi _{1}}}) D_{1} e_{y} = \lambda \) and

$$ {\varvec{ \pi } A_{2} e = (\sum _{i=0}^{S+1} {\varvec{\pi _{1}}}} ) \wedge e_{(yml)}= \mu . $$

Since \(\pi A_{0} e = \lambda \) and \(\pi A_{2} e = \mu ,\) then the queueing inventory model under study is stable if and only if

$$ \lambda < \mu $$

3.2 Stationary Distribution

According to Stewart [11], we can obtain the stationary distribution of the Markov chain under study by solving the set of Eqs. 1 and 2.

$$\begin{aligned} \mathbf{X} Q = 0 \end{aligned}$$
(1)
$$\begin{aligned} { \mathbf{X}} e = 1. \end{aligned}$$
(2)

Let \( { \mathbf{X}} \) be decomposed with Q as following :

$$\begin{aligned}&\qquad \mathbf{X} = ( \mathbf{X}_{0}, \mathbf{X}_{1},\dots )\text { where } \mathbf{X}_{0} =( { \mathbf{X}}_{01},\mathbf{X}_{02}, \dots , { \mathbf{X}}_{0S}); \\&\mathbf{X}_{0k} =( \mathbf{X}_{0k1}, \mathbf{X}_{0k2},\mathbf{X}_{0k3}, \dots , \mathbf{X}_{0kl} ) \text { for } k = 1,2,3, \cdots , S ;\\&\mathbf{X}_{0kr} =( x_{0kr1}, x_{0kr2},x_{0kr3}, \dots , x_{0kry} )\text { for } r = 1,2,3, \cdots , l ;\\&\mathbf{X}_{i} =( { \mathbf{X}}_{i1},\mathbf{X}_{i2}, \dots , { \mathbf{X}}_{iS})\text { for } i = 1,2,3, \cdots ;\\&\mathbf{X}_{ik} =({ \mathbf{X}}_{ik1}, { \mathbf{X}}_{ik2},\mathbf{X}_{ik3}, \dots , { \mathbf{X}}_{ikl});\\&\mathbf{X}_{ikr} =( \mathbf{X}_{ikr1}, \mathbf{X}_{ikr2},\mathbf{X}_{ikr3}, \dots , \mathbf{X}_{ikrm} );\\&\mathbf{X}_{ikrj} =( x_{ikrj1}, x_{ikrj2},x_{ikrj3}, \dots , x_{ikrjy} )\text { for } j = 1,2,3, \cdots , m . \end{aligned}$$

From Eq. 1, we get set of equations as following.

$$\begin{aligned} \mathbf{X}_{0} B_{00} + \mathbf{X}_{1} B_{10}=0; \end{aligned}$$
(3)
$$\begin{aligned} \mathbf{X}_{0} B_{01} + \mathbf{X}_{1} A_{1} + \mathbf{X}_{2} A_{2} = 0; \end{aligned}$$
(4)
$$\vdots $$
$$ \mathbf{X}_{i-1} A_{0} + \mathbf{X}_{i} A_{1} + \mathbf{X}_{i+1}A_{2} = 0 \text{ for } i \ge 2 .$$

\( \text{ where } i \text{ is } \text{ a } \text{ positive } \text{ integer } \text{ number }.\)

There exists a constant matrix R such that

$$\begin{aligned} \mathbf{X}_{i} = \mathbf{X}_{i-1} R \text{ for } i \ge 2 . \end{aligned}$$
(5)

We can rewrite the Eq. 5 as following

$$ \mathbf{X}_{i}=\mathbf{X}_{1} R^{i-1} \text{ for } i \ge 2.$$

We can use the matrix quadratic Eq. 6 to obtain the matrix R.

$$\begin{aligned} R^{2} A_{2} + R A_{1}+A_{0}=0. \end{aligned}$$
(6)

The matrix R can be obtained from \( R_{k+1} = -V - R^{2}_{k} W \) and \(R_{0} = 0\); where \( V = A_{0} A_{0}^{-1}\) and \(W = A_{2} A_{1}^{-1}\). Then, we can find \(X_{0}\) and \(X_{1}\) by solving Eqs. 3 and 4. After that, we must normalize \(X_{0}\) and \(X_{1}\) by using the normalizing condition \( \mathbf{X_{0}} + \mathbf{X_{1}} (I - R)^{-1} e = 1 .\) Then, we use \( \mathbf{X}_{i}=\mathbf{X}_{1} R^{i-1} \text{ for } i = 2,3,\dots .\)

4 Performance Measures

Under steady state, some performance measures of this queueing inventory model can be obtained as following:

  1. 1.

    Expected number of customers in the system

    $$ E[N]= \sum _{i=0}^{\infty } i \mathbf{X}_{i} e. $$
  2. 2.

    Expected number of items in inventory.

    $$ E[I]= \sum _{i=0}^{\infty } \sum _{k=1}^{S} k \mathbf{X}_{ik} e. $$
  3. 3.

    Probability that the server is idle

    $$ b_{0}= \sum _{k=1}^{S} \mathbf{X}_{0k} e .$$

5 Numerical Example

For the arrival process, we consider Markovian arrival process (MAP) with representation \(( D_{0},D_{1})\) of order \(y = 3,\) where

\( D_{0}= \left( \begin{array}{ccc} -8 &{} 1.5 &{} 1 \\ 1.5 &{} -6 &{} 1.5 \\ 1&{} 1 &{} -7 \end{array} \right) \) and \( D_{1}= \left( \begin{array}{ccc} 1.5 &{} 1.5 &{} 2.5 \\ 0.5 &{} 1 &{} 1.5 \\ 2.5&{} 1.5 &{} 1 \end{array} \right) \).

For the service process, we consider PH-representation \( (\beta , T) \) of order \( m=3\), where

\(\beta = \left( \begin{array}{ccc} 0.2,0.5,0.3 \end{array} \right) \),

\( T= \left( \begin{array}{ccc} -12 &{} 3 &{} 4 \\ 6 &{} -13 &{} 3 \\ 5&{} 3 &{} -14 \end{array} \right) \) and \( T^{0}= - T e = \left( \begin{array}{c} 5 \\ 4 \\ 6 \end{array} \right) \).

For the common life time (CLT) for inventoried items, we consider PH-representation \( (\alpha , W) \) of order \( l=3\), where

\(\alpha = \left( \begin{array}{ccc} 0.3,0.4,0.3 \end{array} \right) \),

\( W= \left( \begin{array}{ccc} -0.35 &{} 0.1 &{} 0.2 \\ 0.3 &{} -0.41 &{} 0.1 \\ 0.3&{} 0.2 &{} -0.52 \end{array} \right) \) and \( W^{0}= - W e = \left( \begin{array}{c} 0.05 \\ 0.01 \\ 0.02 \end{array} \right) \).

We fix \( \text{ the } \text{ rate } \text{ of } \text{ lead } \text{ time } \theta = 0.6\) and \( s =3.\)

Now, we analyze the effect of S on the performance measures of the system in the Table 2.

Fig. 1.
figure 1

Effect of S on expected number of customers

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Table 2. Effect of S on various performance measures
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Fig. 2.
figure 2

Effect of S on expected number of items in inventory

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Fig. 3.
figure 3

Effect of S on probability that the server is idle

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From Figs. 1, 2 and 3, we can realize the effect of S on performance measures as following:

  1. 1.

    The expected number of customers in the system E[N] has no change when the maximum inventory level S increases.

  2. 2.

    The expected number of items in inventory E[I] increases when the maximum inventory level S increases.

  3. 3.

    The probability that the server is idle \( b_{0}\) has no change when the maximum inventory level S increases.

6 Conclusion

In this paper, we analyse an MAP/PH/1 queueing inventory model under (s, S) policy with lead time positive and with common life time for inventoried items. In the case of expiry of the common life time for the inventoried items, the supply of items is immediately in local purchase to bring the inventory level to the maximum inventory level S. Different performance measures are estimated under Steady state condition. In this paper, we study the effect of maximum inventory level S on the performance measures of the system numerically. We realize that firstly, the expected number of customers in the system E[N] has no change when the maximum inventory level S increases. Secondly, the expected number of items in inventory E[I] increases when the maximum inventory level S increases. Finally, the probability that the server is idle \( b_{0}\) has no change when the maximum inventory level S increases.