A finite source perishable inventory system with second optional service and server interruptions

In this article, a service facility inventory system with server interruptions and a finite number of sources are considered. The inventory is replenished according to (s, S) ordering policy. Using the matrix methods, the stationary distribution of the stock level, server status and waiting area level is obtained in the steady state case. The Laplace-Stieltjes transform of the waiting time of the tagged customer is derived. Many impartment system performance measures are derived and the total expected cost rate is computed under a suitable cost structure. The results are illustrated numerically.


Introduction
Schwarz et al. [18] introduced the idea of inventory with positive service time.They have assumed that when the stock level is empty, each arriving customer enters into the queue.Berman et al. [2] derived deterministic approximations for a queueing-inventory system with a service facility.Berman & Kim [3] analyzed a single server inventory model with the assumption of instantaneous replenishment and a service facility.Berman & Sapna [5] considered a queueing-inventory model with single server and finite capacity.They assumed that customer arrival follows Poisson process, arbitrarily distributed service times and zero lead times.Berman & Kim [4] addressed an infinite capacity queueing -inventory system with Poisson arrivals, exponential distributed lead times and service times.The authors identified a reorder policy that maximized the system proceeds.Krishnamoorthy & Anbazhagan [13] studied a queueing-inventory system with N policy and the finite waiting hall.They have assumed that if the customer level reaches a prefixed level N , then the server starts service immediately.Otherwise, server does not provide service to the waiting customers.Two other papers where a service facility inventory model is considered, are by Krishnamoorthy et al. [14,15].The paper [14] is analyzed for a queueing-inventory model with instantaneous replenishment and the service process is subject to interruptions.The discussion in [15] is a retrial queueing-inventory model with positive lead-time and the service process is subject to interruptions.Jeganathan & Periyasamy [11] studied a perishable inventory system with repeated attempts and the service process is subject to interruptions.They analyzed the system with the restriction that both the waiting area and orbit size are finite.Jeganathan et al. [9] discussed a service facility inventory system with multiple server vacations and server is subject to interruptions.For a detailed study in the service facility inventory models with an infinite number of sources the reader is referred to [1,6,8,10,12,22].An (s, S) inventory system with finite source was first initiated by Sivakumar [20].He assumed that lifetime of each commodity is exponential and lead-time distribution is exponential.The author considered the constant retrial policy i.e., the probability of retrial is independent of the number of demands in the orbit.Shophia Lawrence et al. [19] considered a service facility inventory management system with the service time, the lead time are assumed to have Phase type distribution and finite source.Yadavalli et al. [23] analysed a service facility retrial inventory model with finite source and multi homogeneous servers.Yadavalli et al. [21] studied a two heterogeneous queueing-inventory system including one server is perfectly reliable and another server is subject to interruptions.In a recent paper, Jeganathan [7] analysed a mixed priority retrial inventory system with additional optional service and finite source.
A finite source queueing-inventory model with server interruptions and extra optional service is motivated by the service facility system with limited customers such as military canteen providing service to soldiers or a company canteen serving the members of the specific working area in the business.In this paper, a continuous review (s, S) inventory model with server interruptions, second optional service and finite source simultaneously are considered.
In the next section, the mathematical model is explained and the notations used in this paper are defined.Model analysis and the steady state analysis are proposed in section 3.In section 4, the waiting time analysis of a customers in the waiting area is discussed.Various important performance measures are derived in section 5.In section 6, the total expected cost rate is derived.In section 7, optimality of the cost function and its sensitivity with respect to various parameters using numerical examples are presented.The last section is meant for conclusion.

Model description
In this work, finite-source queueing-inventory models with the following assumption are studied.Consider a service facility wherein perishable items are stored and the items are distributed to the demanding customers.The maximum stock level is S. The customers are generated by a finite number of identical sources N , (1 < N < ∞) and the demand time points form a quasi-random distribution with rate λ(> 0).An arriving customer finds the system either when the empty stock level or the server is busy or the server is on interruptions, then with probability r he/she enters into the waiting area.Otherwise, he/she balks (do not join) with probability 1 − r.There is a single waiting area for the customers and the demand is for single item per customer.Before distributing items to the demanding customers, some primary service on the item is given first.In this article this type of service is referred to as first essential service (FES).This system has a single server who gives preliminary FES indicated by the rate µ 1 to all arriving customers one by one according to FIFO (first in first out) discipline.When the FES of a customer is completed, the server may offer a second optional service (SOS) indicated by the rate µ 2 with probability p to only those customers who opt for it otherwise leaves the system with the complementary probability q, where p+q=1.While the server is in working state it may be interrupted at any time with interruption rate α 1 during first essential service (FES) and α 2 during second optional service (SOS).When the server interruption occurs, it is immediately sent for repairing where repair time indicated by η 1 for FES and η 2 for SOS.After repairing, the server provides residual service of the customers of both of the phases (FES or SOS).It is assumed that if the server is in interruption, no more interruption can be caused on the server.The service times of the FES and SOS, and the interruption times are assumed to follow an exponential distribution.
The operating policy is (s, S) policy with exponential lead times for the ordered items.According to the ordering policy, when the stock level downfall to s, an order for Q(= S −s > s+1) items are placed.Lifetime of each item has negative exponential distribution with rate γ > 0. The positive lead-time of the replenishment is assumed to be exponential with the rate β(> 0).All stochastic processes involved in the system are independent of each other.The notation used in this paper follows below.e : a column vector of appropriate dimension containing all ones 0 : zero matrix [A] ij : entry at (i, j) th position of a matrix A
• Finally, the intensity of passage for the state (i 1 , i 2 , i 3 ) is given by − Hence, we have d Define the following ordered sets: By ordering the sets of state space as (≪ 0 ≫, ≪ 1 ≫, ≪ 2 ≫, . . ., ≪ S ≫), the infinitesimal generator Θ can be conveniently expressed in a block partitioned matrix with entries otherwise.
Step 1. Solve the following system of equations to find the value of Φ Step 2. Compute the values of Step 3. Using, Step 1 and Step 2, calculate the value of Φ (i 1 ) , i 1 = 0, 1, . . ., S. That is,

Waiting time analysis
In this section, the aim is to derive the waiting time for the customer.The specific as the time between the arrival times of the customer and immediate upon which he gets service.We will symbolize this continuous time random variable as W .The aim is to derive the probability distribution of W and to derive n th order moments of W .Note that W is zero when the is in the state (i 1 , 0, 0), i 1 ∈ V S 1 .Consequently, the probability that the customer does not have to wait is given by To obtain the distribution of W , some auxiliary variables are defined.Let us consider the Markov process at an arbitrary time t and assume that the system in the state (i 1 , i 2 , i 3 ), i 3 > 0. We tag any of those waiting customer and W (i 1 ,i 2 ,i 3 ) denotes the time until the selected customer gets the desired item.Let W * (y) = E[e −yW ] and W * (i 1 ,i 2 ,i 3 ) (y) = E[e −yW (i 1 ,i 2 ,i 3 ) ] respectively, denote the unconditional and conditional waiting time.Then, we have φ (0,0,i 3 ) W * (0,0,i 3 +1) (y) ( To derive W * (i 1 ,i 2 ,i 3 ) , we introduce an auxiliary Markov chain on the state space where { * } represents an absorbing state.The chain is on a state (i 1 , i 2 , i 3 ), we apply a first-step argument in the auxiliary chain to resolve W * (i 1 ,i 2 ,i 3 ) (y).Then (see [16], Theorem 6.21) the functions W * (i 1 ,i 2 ,i 3 ) (y), (i 1 , i 2 , i 3 ) ∈ E are the smallest non-negative solution to the system For i 2) where where Using the linear equations ( 2)-( 4), we can compute the values of W * (y) for a given y and also we can utilize the system of linear equations to obtain a recursive algorithm for calculating moments for the waiting times.By differentiating (n+1) times ( 2)-(4) the system of linear equations, and evaluating at y = 0, we arrive at For where where where Equations ( 5)-( 7) are used to determine the unknowns E W (i 1 ,i 2 ,i 3 ) , (i 1 , i 2 , i 3 ) ∈ E in terms of the moments of one order less.Noticing that E W (n) (i 1 ,i 2 ,i 3 ,) = 1, for n = 0, we can obtain the moments up to a desired order in a recursive way.For determine the moments of W we differentiate W * (y) and evaluate at y = 0, we have which provides the n th moments of the unconditional waiting time in terms of conditional moments of the same order.

System performance measures
In this section, some measures of system performance in the steady state are derived.Using this, the total expected cost rate is derived.

Expected inventory level
Let η I denote the excepted inventory level in the steady state, then

Expected reorder rate
Let η R denote the expected reorder rate in the steady state.A reorder is placed when the inventory level drops from s + 1 to s.This may occur in the following three cases: • The server completes a first essential service for the customer.
• Any one of the s items fails when the server is busy/interruption during FES.
• Any one of the (s + 1) items fails when the server is idle/busy/interruption during SOS.

Expected number of customers in the waiting area
Let Γ 1 denote the expected number of customers in the steady state, then

Expected waiting time
Let η W denote the expected waiting time of the customers in the waiting area.Then by Little's formula where Γ 1 is the expected number of customers in the waiting area and the effective arrival rate of the customer [17], Γ 2 is given by (N − i 3 )λφ (i 1 ,0,0)

Effective interruption rate
Let η IN T R denote the effective interruption rate which is given by α 2 φ (i 1 ,2,i 3 ) .

Effective repair rate
Let η RR denote the effective repair rate which is given by η 2 φ (i 1 ,4,i 3 ) .

Probability that server is idle
Let η P I denote the probability that server is idle is given by φ (i 1 ,0,0) .

Probability that server is working
Let η P W denote the probability that server is working is given by (φ (i 1 ,2,i 3 ) + φ (i 1 ,4,i 3 ) ).

Probability that server is on FES
Let η P F ES denote the probability that server is providing FES is given by 4.12 Probability that server is on SOS Let η SOS denote the probability that server is providing SOS is given by φ (i 1 ,2,i 3 ) .
6 Numerical illustrations In this section, some numerical examples that reveal the possible convexity of the total expected cost rate are discussed.A typical 3-dimensional plot of T C(S, s) is presented in Figure 1.The numerical search procedure is employed to obtain the optimal values of S, s and T C (say S * , s * and T C * ). the effect of varying the cost and other system parameters on the optimal values and the results agreed with what one would expect, have been studied.Some of the results are presented in Tables 4 through 11 where the lower entry in each cell gives the optimal expected cost rate and the upper entries the corresponding S * and s * .
Example 1 First, the behaviour of the cost function is explored by considering as the function of two variables by fixing the others at a constant level.Tables 1 − 3

Summary and conclusion
In this article, a continuous review stochastic queueing-inventory system with (s, S) control policy, server interruptions and finite source was analyzed.The model is analyzed within the framework of Markov processes.Stationary distribution of the number of customers in the waiting area, the server status and the inventory level is obtained in the steady state.
Various system performance measures are derived and the long-run total expected cost rate is derived.The waiting time distribution is derived.A sensitivity analysis is numerically performed on the expected total cost function with respect to various parameters of the model.The authors are working in the direction of MAP (Markovian arrival process) arrival for the customers and service times follow PH-distributions.

Figure 1 :
Figure 1: A three dimensional plot of the cost function T C(s, S)

Figure 4 :λ
Figure 4: T C versus β for different values of µ 1

Figure 5 :β
Figure 5: T C versus γ for different values of µ 1

Figure 7 :
Figure 7: η W versus λ for different values of N

Figure 9 :λ 5 Figure 10 :λ 5 Figure 11 :
Figure 9: η W versus η 1 for different values of η 2 All the costs and other parameters are assigned fixed values which are indicated in each Table.The value that is shown bold is the least among the values in that row and the value that is shown underlined is the least in that column.It may be observed that, these values in each Table exhibit a (possibly) local minimum of the function of the two variables.Also it may be observed that, the total expected cost rate function T C(S, s, N ) is more sensitive to changes in N than to changes in S and s.
, give the total expected cost rate as a function of T C(S, s, 10), T C(50, s, N ) and T C(S, 7, N ).