Analysis of a Multiserver Queueing-Inventory System

We attempt to derive the steady-state distribution of theM/M/c queueing-inventory system with positive service time. First we analyze the case of c = 2 servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the (s, Q) 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 (s, Q) and the corresponding expected minimum cost are computed. As in the case ofM/M/c retrial queue with c ≥ 3, we conjecture thatM/M/c for c ≥ 3, queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach.Wederive 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 s to s transitions of the inventory level (i.e., the first return time to s) is computed. We also obtain several system performance measures.


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 queueinginventory 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 steadystate distribution of an //1/∞ 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  + 1 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 //2 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 3 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 .
We have Then using (3) we get the components of the vector  explicitly as Thus, we have the following lemma for stability of the system under study.
Lemma 1.The stability condition of the //2 queueinginventory 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  0 e <  2 e for the Markov chain to be stable.With a bit of algebra, this simplifies to  < [2− 0 /(1− 0 )].
For future reference we define  1 as

Computation of the Steady-State Probability
For computing the steady-state probability vector of the process {X() |  ≥ 0}, we first consider a queueing-inventory system with unlimited supply of inventory items (i.e., classical //2 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 {N() |  ≥ 0}, where N() is the number of customers in the system at time .Its infinitesimal generator G 1 is given by ] .
(9) Let  be the steady-state probability vector of G 1 .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 {I() |  ≥ 0}.Here I() is the inventory level at time .Its infinitesimal generator G 2 is given by Let  = ( 0 ,  1 , . . .,   ) be the steady-state probability vector of the process {I() |  ≥ 0}.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 x be the steady-state probability vector of the original system.Then the steady-state vector must satisfy the set of equations Partition x by levels as where the subvectors of x are further partitioned as Then by using the relation xW 1 = 0, we get We assume a solution of the form for constants Θ   , and then verify that the system of equations given in ( 16) is satisfied.
The constants Θ   's are given by where where

Advances in Operations Research
Consider where If we note xe = 1 and (20) we have Write  = 1 + (/)(( + )/)  .Then dividing each Θ       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  1 < 1 holds.Then the components of the steady-state probability vector of the process {X() |  ≥ 0} with generator matrix W 1 are   () =  −1 Θ       ,  ≥ 0, 0 ≤  ≤ , 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 onedimensional objects one of which corresponds to the number of customers in an //2 queue and the other to the number of items in the inventory.

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   loss =  loss /  .(x) Effective arrival rate is as follows: (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:

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 //2 queueing-inventory model we introduce the cost function F(2, , ) defined by where  is fixed cost for placing an order,  1 is the cost incurred due to loss per customer,  2 is waiting cost per unit time per customer during the stock out period,  3 is variable procurement cost per item,  4 is the cost incurred per busy server,  5 is the cost incurred per idle server, and ℎ is unit holding cost of inventory per unit per unit time.We assign the following values to the parameters:  = 5,  = 3,  = 1,  = $500,  1 = $100,  2 = $50,  3 = $25,  4 = $10,  5 = $20, and ℎ = $2.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.

𝑀/𝑀/𝑐 (𝑐 ≥ 3) Queueing-Inventory System
Next we consider // queueing-inventory system with positive service time for 3 ≤  ≤ .We keep the model assumptions the same as in Section 2. Hence the service rate is , for  varying from 0 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 .

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  =  0 +  1 +  2 is the infinitesimal generator of the finite state CTMC on the set {0, 1, . . ., }.Write  as ( 0 ,  1 , . . .,   ).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
Proof.The proof is on the same lines as that of Lemma 1.

Advances in Operations Research
Next we compute the steady-state probability vector of W 2 under the stability condition.Let y denote the steadystate probability vector of the generator W 2 .So y must satisfy the relations Let us partition y by levels as y = (y 0 , y 1 , y 2 , . ..), where the subvectors of y are further partitioned as The steady-state probability vector y is obtained as where  is the minimal nonnegative solution to the matrix quadratic equation and the vectors y 0 , y 1 , . . ., y c−1 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 .

Conditional Probability Distributions
We could arrive at an analytical expression for system state probabilities of //2 queueing-inventory system.However for the // queueing-inventory system with  ≥ 3, 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.

Conditional Probability Distribution of the Inventory Level
Conditioned on the Number of Customers in the System.Let  = ( 0 ,  1 , . . .,   ) 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.

Advances in Operations Research
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 Γ 1 be the infinitesimal generator of the corresponding Markov chain.
The infinitesimal generator Γ 1 is given by and the inventory level distribution  can be obtained from the equations Γ 1 = 0 and e = 1, and we get where (ii) Case of  ≥ .
The infinitesimal generator Γ 2 is given by ) .

Conditional Probability Distribution of the Number of
Customers Given the Number of Items in the Inventory.Let   ,  ≥ 0, denote the probability that there are  customers in the system conditioned on the inventory level at .We have three different cases.
(ii) When 0 <  < , ] 0 () ,  = 0,   + ( − 1)  + 1 ]  () ,  ≥ . (63) The first term on the right hand side of the case of 1 ≤  ≤  in (63) has two factors; the former represent probability of an arrival before service completion as well as replenishment when there were −1 customers and  inventory in the system.Similar explanations stand for the remaining terms and also for other expressions for   .

Performance Measures
iv) Mean number of busy servers is as follows: (vii) Mean number of replenishments per time unit is   = (∑ ∞ =0 ∑  =0   ()).(viii) Mean number of departures per unit time is as follows:  (xiv) Mean number of customers waiting in the system when inventory is available is W = ∑ ∞ =1 ∑ + =1   ().(xv) Mean number of customers waiting in the system during the stock out period is W = ∑ ∞ =1   (0).

Analysis of Inventory Cycle Time
We define the inventory cycle time random variable, Γ cycle , as the time interval between two consecutive instants at which the inventory level drops to .Thus Γ cycle is a random variable whose distribution depends on the number of customers at the time when inventory level dropped to  at the beginning of the cycle and the inventory level process prior to replenishment.We proceed with the assumption that  = 1.If the number of customers present in the system is at least  +  when the order for replenishment is placed, then we need not have to look at future arrivals to get a nice form for the cycle time distribution.In fact it is sufficient that there are at least  customers at that epoch.However in this case the service rate during lead time may drop below  even when there are at least  items in the inventory.This is so since number of customers may go below .Thus we look at various possibilities below.is not unique since the service rate strongly depends on both inventory level and number of customers in the system.The case of ℓ < : here again the procedure is similar to that corresponding to ℓ ≥ , but less than  + .The initial state is (, ℓ).After exactly  service completions with a replenishment within this cycle and with arrivals truncated at that epoch which ensure rate  for as many services as possible, the absorption state of the Markov chain generated corresponds to a departure epoch with  items in the inventory.Here again the cycle time has a PH distribution with representation which is not unique because the service rates may change depending on the number of customers in the system and the number of items in the inventory.

Optimization Problem II
We look for the optimal pair of control variables in the model discussed above.Now for computing the minimal cost of (, ) model we introduce the cost function: F(, , ), which is defined by where  = 40,  = 81, and ,  1 ,  2 ,  3 ,  4 ,  5 , and ℎ are the same input parameters as described in Section 4. We provide optimal  and corresponding minimum cost for various  values.From Table 2 we notice that the optimal value of  is 6 for various  values, presumably because of the high holding cost.
In Table 3, we examine the optimal pair (, ) and the corresponding minimum cost for various  and , keeping other parameters fixed (as in Section 4).

Conclusions
In this paper we studied multiserver queueing-inventory system with positive service time.First we considered two server queueing-inventory systems, where the steady-state distributions are obtained in product form.Further, we analyse queueing-inventory system with more than two servers.As observed in [21] by Falin and Templeton, we conjecture that // for  ≥ 3, queueing-inventory system, does not have analytical solution.So such cases are analyzed by algorithmic approach.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.We also provided the cycle time distribution of two consecutive  to  transitions of the inventory level (i.e., the first return time

(
xi) Mean sojourn time of the customers in the system is   =   /  .(xii) Mean waiting time of a customer in the queue is   =   /  .

(
xii) Mean sojourn time of the customers in the system is   =   /  .(xiii) Mean waiting time of a customer in the queue is   =   /  .
Expected loss rate of customers when the inventory level is zero per cycle is   loss =  loss /  .(xi) Mean number of customers arriving per unit time is as follows:

Table 2 :
Optimal server  and minimum cost.