#### Abstract

We attempt to derive the steady-state distribution of the queueing-inventory system with positive service time. First we analyze the case of servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair and the corresponding expected minimum cost are computed. As in the case of retrial queue with , we conjecture that for , queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutive to transitions of the inventory level (i.e., the first return time to ) is computed. We also obtain several system performance measures.

#### 1. Introduction

The notion of inventory with positive service time was first introduced by Sigman and Simchi-Levi [1]. They assumed arbitrarily distributed service time, exponentially distributed replenishment lead time with customer arrival forming a Poisson process. Under the condition of stability of the system, they investigate several performance characteristics. In the context of arbitrarily distributed lead time the readers attention is invited to a very recent paper by Saffari et al. [2] where the authors provide a product form solution for system state probability distribution under the assumption that* no customer joins the system when inventory level is zero*.

Reference [1] by Sigman and Simchi-Levi was followed by [3] of Berman et al. with deterministic service time wherein they formulated the model as a dynamic programming problem. A review paper by Krishnamoorthy et al. [4] provides the details of the research developments on queueing theory with positive service time. Schwarz et al. [5] were the first to produce product form solutions for single server queueing-inventory problem with exponentially distributed service time as well as lead time and Poisson input of customers. They arrived at product form solution for the system state distribution. Nevertheless this is achieved under the assumption that customers do not join when the inventory level is zero (of course, [2] of Saffari et al. is the extension of this to arbitrary distributed lead time). This is despite the strong correlation between the lead time and the number of customers joining the system during that time. Subsequently several authors made the above assumption in their investigations to come up with product form solution, the details of which could be seen below. Krishnamoorthy and Viswanath [6] subsume Schwarz et al. [5] by extending the latter to production inventory with positive service time. References [7] of Sivakumar and Arivarignan, [8] of Krishnamoorthy and Narayanan, [9] of Deepak et al., [10] of Schwarz and Daduna, [11] of Schwarz et al., and [12] of Krenzler and Daduna are a few other significant contributions to inventory with positive service time. Protection of production and service stages in a queueing-inventory model, with Erlang distributed service and interproduction time, is analyzed by Krishnamoorthy et al. [13].

Classical queue with inventoried items for service is also studied by Saffari et al. [14] where the control policy followed is and lead time is mixed exponential distribution. Customers arriving during zero inventory are lost forever. This leads to a product form solution for the system sate probability. Schwarz et al. [11] consider queueing networks with attached inventory. They consider rerouting of customers served out from a particular station when the immediately following station has zero inventory. Thus no customer is lost to the system. The authors derive joint stationary distribution of queue length and on-hand inventory at various stations in explicit product form. A recent contribution of interest to inventory with positive service time involving a random environment is by Krenzler and Daduna [15] wherein also a stochastic decomposition of the system is established. They prove a necessary and sufficient condition for a product form steady-state distribution of the joint queueing-environment process to exist. A still more recent paper by Krenzler and Daduna [12] investigates inventory with positive service time in a random environment embedded in a Markov chain. They provide a counter example to show that the steady-state distribution of an system with policy and lost sales need not have a product form. Nevertheless, in general, loss systems in a random environment have a product form steady-state distribution. They also introduce a blocking set where all activities other than replenishment stay suspended whenever the Markov chain is in that set. This resulted in arriving at a product form solution to the system state distribution.

The work on multiserver queueing-inventory systems is scarce. Nair et al. [16] consider an inventory system with number of servers varying from to , depending on the inventory position. Another contribution is by Yadavalli et al. [17] wherein the authors consider a finite customer source system (this paper contains a few additional references to multiserver inventory system).

In all work quoted above, customers are provided an item from the inventory on completion of service. Nevertheless, there are several situations where a customer may not be served/may not purchase the item with probability one at the end of his service. For example, customers who may buy an item arrive at a retail shop where there are one or more (finite number) servers (sales executives). The servers explain to each customer the features of product. The time required for this may be regarded as the service time. After listening to the server each customer, independently of the others, decides whether to buy the item (probability ) or leaves the system without purchasing the item. A less realistic example is as follows: a candidate appears for an interview against a position. At the end of the interview the candidate decides to accept the offer of job with probability and with complementary probability rejects it. In this case the job is taken as an inventory. In this connection one may refer to Krishnamoorthy et al. [18] for some recent developments.

We arrange the presentation of this paper as indicated below: in Section 2 the queueing-inventory problem is mathematically formulated. The product form solution of the steady-state probability distribution, including some important performance measures, is obtained in Section 3. Further we numerically investigate the optimal pair values and the minimal cost for different values of . Section 5 discusses the with (greater than or equal to but less than ) queueing-inventory problems by using algorithmic approach. Section 6 gives some conditional probability distributions and a few performance measures for the (3) server case. Section 7 analyzes the distribution of the inventory cycle time. In Section 8 the optimal and the corresponding minimal cost for different values of are investigated. Further we look for the optimal pair values that would result in cost minimization for different pairs of values of and .

#### 2. Mathematical Modelling of the Queueing-Inventory Problem

First we consider an queueing-inventory system with positive service time. Customer arrival process is assumed to be Poisson with rate . Each customer requires a single item having random duration of service which follows exponential distribution with parameter . However, it is not essential that inventory is provided to the customer at the end of his service. More precisely, the item is served with probability at the end of a service or else it is not provided with probability . A crucial assumption of this model is that customers do not join the system when the inventory level is zero. When the number of customers is at least two and not less than two items are in inventory, the service rate is . When the inventory level reaches a prespecified value , a replenishment order is placed for units with . We fix as the maximum number of items that could be held in the system at any given time. The lead time follows exponential distribution with parameter . Then is a CTMC with state space , where is called the th level. In each of the levels the number of items in the inventory can be anything from to . Accordingly we write . The infinitesimal generator of this CTMC is where contains transition rates within ; represents the transitions from level 1 to level 0; contains the transitions within level 1; represents the transition from level to level ,â€‰â€‰; represents the transitions within for ; and represents transitions from to ,â€‰â€‰. The transition rates are Note that all entries (block matrices) in are of the same order, namely, , and these matrices contain transition rates within level (in the case of diagonal entries) and between levels (in the case of off-diagonal entries).

##### 2.1. Analysis of the System

In this section we perform the steady-state analysis of the queueing-inventory model under study by first establishing the stability condition of the queueing-inventory system. Define . This is the infinitesimal generator of the finite state CTMC corresponding to the inventory level for any level (1). Let denote the steady-state probability vector of . That is, Write We have Then using (3) we get the components of the vector explicitly as Since the Markov chain is an LIQBD, it is stable if and only if the left drift rate exceeds the right drift rate. That is, Thus, we have the following lemma for stability of the system under study.

Lemma 1. *The stability condition of the queueing-inventory system under consideration is given by .*

*Proof. *From the well-known result by Neuts [19] on the positive recurrence of the Markov chain associated with , we have for the Markov chain to be stable. With a bit of algebra, this simplifies to .

For future reference we define as

#### 3. Computation of the Steady-State Probability

For computing the steady-state probability vector of the process , we first consider a queueing-inventory system with unlimited supply of inventory items (i.e., classical queueing system). The rest of the assumptions such as those on the arrival process and lead time are the same as given earlier. Designate the Markov chain so obtained as , where is the number of customers in the system at time . Its infinitesimal generator is given by Let be the steady-state probability vector of . Partitioning by levels we write as Then the steady-state vector must satisfy From the relation (11) we get the vector explicitly as follows: Further we consider an inventory system with negligible service time and no backlog of demands. The assumptions such as those on the arrival process and lead time are the same as given in the description of the model. Denote this Markov chain by . Here is the inventory level at time . Its infinitesimal generator is given by

Let be the steady-state probability vector of the process . Then satisfies the relations That is, at arbitrary epochs the inventory level distribution is given by Using the components of the probability vector , we will find the steady-state probability vector of the original system. Let be the steady-state probability vector of the original system. Then the steady-state vector must satisfy the set of equations Partition by levels as where the subvectors of are further partitioned as Then by using the relation , we get We assume a solution of the form for constants , and then verify that the system of equations given in (16) is satisfied.

The constants â€™s are given by where .

Consider where .

Consider where ,â€‰â€‰.

Thus we have If we note and (20) we have Write . Then dividing each by we get the steady-state probability vector of the original system.

Thus we arrive at our main theorem.

Theorem 2. *Suppose that the condition holds. Then the components of the steady-state probability vector of the process with generator matrix are ,â€‰â€‰,â€‰â€‰, the probabilities correspond to the distribution of number of customers in the system as given in (12), and the probabilities are obtained in (15).*

The consequence of Theorem 2 is that the two-dimensional system can be decomposed into two distinct one-dimensional objects one of which corresponds to the number of customers in an queue and the other to the number of items in the inventory.

##### 3.1. Performance Measures

(i)Mean number of customers in the system is as follows: (ii)Mean number of customers in the queue is as follows: (iii)Mean inventory level in the system is as follows: (iv)Mean number of busy servers is as follows: (v)Depletion rate of inventory is as follows: (vi)Mean number of replenishments per time unit is as follows: (vii)Mean number of departures per unit time is as follows: (viii)Expected loss rate of customers is as follows: (ix)Expected loss rate of customers when the inventory level is zero per cycle is .(x)Effective arrival rate is as follows: (xi)Mean sojourn time of the customers in the system is .(xii)Mean waiting time of a customer in the queue is .(xiii)Mean number of customers waiting in the system when inventory is available is as follows: (xiv)Mean number of customers waiting in the system during the stock out period is as follows:

#### 4. Optimization Problem I

In this section we provide the optimal values of the inventory level and the fixed order quantity . Now for computing the minimal costs of queueing-inventory model we introduce the cost function defined by where is fixed cost for placing an order, is the cost incurred due to loss per customer, is waiting cost per unit time per customer during the stock out period, is variable procurement cost per item, is the cost incurred per busy server, is the cost incurred per idle server, and is unit holding cost of inventory per unit per unit time. We assign the following values to the parameters: ,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰,â€‰â€‰and . Using MATLAB program we computed the optimal pairs and also the corresponding minimum cost (in Dollars). Here is varied from 0.1 to 1 each time increasing it by 0.1 unit. The optimal pair and the corresponding cost (minimum) are given in Table 1.

#### 5. Queueing-Inventory System

Next we consider queueing-inventory system with positive service time for . We keep the model assumptions the same as in Section 2. Hence the service rate is , for varying from to , depending on the availability of the inventory and customers. When the number of customers is at least and not less than items are in the inventory, the service rate is . Write . Then is a CTMC with state space , where is the collection of states as defined in Section 2. The infinitesimal generator of the CTMC is and the transition rates are

For ,

##### 5.1. System Stability and Computation of Steady-State Probability Vector

The Markov chain under consideration is a LIQBD process. For this chain to be stable it is necessary and sufficient that where is the unique nonnegative vector satisfying and â€‰â€‰+â€‰â€‰â€‰â€‰+â€‰â€‰ is the infinitesimal generator of the finite state CTMC on the set . Write as . Then we get from (42) the components of the probability vector explicitly as From the relation (41) we have the following.

Lemma 3. *The stability condition of the queueing-inventory system under study is given by , where .*

*Proof. *The proof is on the same lines as that of Lemma 1.

Next we compute the steady-state probability vector of under the stability condition. Let denote the steady-state probability vector of the generator . So must satisfy the relations Let us partition by levels as where the subvectors of are further partitioned as The steady-state probability vector is obtained as where is the minimal nonnegative solution to the matrix quadratic equation and the vectors can be obtained by solving the following equations: Now from (49), we get where subject to normalizing condition Since cannot be computed explicitly we explore the possibility of algorithmic computation. Thus, one can use logarithmic reduction algorithm as given by Latouche and Ramaswami [20] for computing . We list here only the main steps involved in logarithmic reduction algorithm for computation of .

*Logarithmic Reduction Algorithm for *

*Step 0*.â€‰â€‰, , , and .

*Step 1*. Consider
Continue Step 1 until .

*Step 2*. .

#### 6. Conditional Probability Distributions

We could arrive at an analytical expression for system state probabilities of queueing-inventory system. However for the queueing-inventory system with , the system state distribution does not seem to have closed form owing to the strong dependence between the inventory level, number of customers, and the number of servers in the system. In this section we provide conditional probabilities of the number of items in the inventory, given the number of customers in the system and also that of the number of customers in the system conditioned on the number of items in the inventory.

##### 6.1. Conditional Probability Distribution of the Inventory Level Conditioned on the Number of Customers in the System

Let be the probability distribution of the inventory level conditioned on the number of customers in the system. Then we get explicit form for the conditional probability distribution of the inventory level conditioned on the number of customers in the system. We formulate the result in the following lemma.

Lemma 4. *Assume that is the number of customers in the system at some point of time. Conditional on this we compute the inventory level distribution where there are items in the inventory. We consider two cases as follows. *(i)*When , the inventory level probability distribution is given by
*(ii)*When , the inventory level probability distribution is derived by
*

*Proof. *Let be the infinitesimal generator of the corresponding Markov chain.(i)Case of .The infinitesimal generator is given byand the inventory level distribution can be obtained from the equations and , and we get
where