An inventory model with self-generation of priorities and retrial of customers

This paper mainly focuses on a single server retrial inventory system. The arrival of customers follows the Poisson process, and when the inventory level depletes to the reorder level s due to service, replenishment order placed according to (s, Q) policy. The lead-time governed by an exponential distribution with parameter β. A customer who finds the server busy or inventory dry moves to an orbit of infinite capacity with probability σ. Customers in orbit independently tries to access the server at a rate of θ, which depends on the number of customers in orbit. A retrial customer returns to the orbit with probability δ if the server is busy or inventory is dry. Each customer in orbit generates priority according to an exponential distribution with rate γ. If the server is free, a priority generated customer will get immediate service. Else such a customer have to wait in a space A 1 of capacity one, which is reserved only for priority generated customers. The service times follows exponential distribution with different rates for ordinary and priority generated customers. The model turned out to be a level-dependent quasi birth-death(LDQBD) process, and we use Matrix-Analytic Method as a tool for obtaining a solution.


Introduction
There are abundance of probability models on priority queues in literature(Gross and Harris [1] chapter 3, Jaiswal [2] chapters 3 and 7, and Takagi [3] chapter 3). Studies on priority queues found many applications in health care systems Brahimi [4], Taylor [5]. Some external priority rules used in all of these models. In many applied areas, this approach is inadequate. The concept of self-generation of priorities of customers in queues is coined by Gomez-Corral, Krishnamoorthy, and Viswanath [6,7]. The preliminary works on self-generation of priorities are done by Krishnamoorthy, S. Babu, and Viswanath [8,9]. The idea of retrial queues are explored extensively in a survey by Yang [10] and Falin [11]. A book by Falin and Templeton [12] discussed all concepts of retrial queues revealed at that time. Along with the concept of self-generation of priorities, S. Babu discussed some models with the retrial of customers [13,14]. Jomy Punalal and S. Babu extend this work with customer induced interruption in [15]. Inventory is a physical stock of items to meet future demands. A voluminous number of deterministic inventory systems can found in Churchman [16], Hadley and Whitin [17], Naddor [18] and Sahin [19]. An inventory model with positive service time can found in the survey [20], a discussion [21] and a comparison of three such models [22] are innovative works of Krishnamoorthy   In this paper, we discuss a retrial inventory model with special emphasize on self-generation of priorities. The first section of the paper contains an accurate description of the model. Mathematical formulation, matrix transitions and system stability discussed in succeeding sections. Some essential system characteristics derived, and we define an expected total profit function with the help of these measures. Different system parameters in profit function and its effect in various performance measures discussed and optimum (s, S) pair also calculated. Results are shown graphically in the last section.

Model Description
The arrival of customers to a retrial inventory follows a Poisson process with parameter λ. When the level of inventory depletes to the reorder level s due to service of the single server, a replenishment order placed according to (s, Q) policy. The order quantity is Q = S − s. The lead-time ruled by an exponential distribution with parameter β. The customer will move to an orbit of infinite capacity with probability σ if the customer finds the busy server or dry inventory. Otherwise, the customer is lost with probability 1 − σ. Customers in orbit independently tries to access the server with rate θ in such a fashion that the rate of a retrial is iθ provided there are i customers in orbit. A retrial customer returns to the orbit with probability δ(< 1) if the server busy or inventory dry. Otherwise, such a customer will be lost forever with probability 1 − δ. Each customer in orbit generates priority according to an exponential distribution with rate γ. If the server is free, the priority generated customer will get immediate service. Else, such customer have to wait in a waiting space A 1 of capacity one, which reserved for priority generated customers. The duration of service times of ordinary customer follows an exponential distribution with parameter µ 1 , and that of priority generated customer follows an exponential distribution with parameter µ 2 . Pictorial representation of the model shown in FIGURE 1.

Stability of System
Theorem: The above discribed system is stable. Proof: To prove the stability, we consider Lyapunov test function defined by φ(r) = i where r is a state in level i. For a state r in level i, the mean drift where r , r , r vary over states belonging to levels i − 1, i and i + 1 respectively. Then when server is idle and inventory not dry, −iγ − iθ(1 − δ) + λσ, otherwise.
As (1 − δ) > 0, for any > 0, we can find K * large enough that y r < − for any r belonging to level i ≥ K * . Thus the theorem follows from Tweedie's result [23].

Stationary Probability vectors
Assuming Q is irreducible and the steady state probability vector of the process χ be ξ = (ξ 0 , ξ 1 , ξ 2 , · · · ). Now by a theorem from Nuets [24] ξ satisfies the relation When the number of retrying customers exceeds a large number, the most of the retrials become unsuccessful and the retrials greater than a large number N will have no effect on the system. The truncation level N is calculated by using Neuts-Rao Truncation [25] we find the steady state distribution ξ 0 of the finite state Markov chain with generator A 10 + R 1 A 21 . Finally each ξ i is divided with ∞ i=0 ξ i e to compute ξ. • Prob(the server is idle)= P idle = ∞ i=0 x i e. • Prob(the server is busy with ordinary customer)= P sbor = ∞ i=0 y i e. • Prob(the server is busy with priority generated customer)= P sbpr = ∞ i=0 z i e. • Prob(the server is idle with customers in the orbit)= P sidco = ∞ i=0 x i e − x 1 e.

Some important system characteristics
• Expected number of customers in the orbit, E or = ∞ i=1 iξ i e. • Expected number of customers in the orbit when server is idle, is a row vector with respect to C(t) = c, N 2 (t) = j and I(t) = k, c = 0, 1, 2; k = 0, 1, 2, · · · S; then • Expected ordinary customers in the orbit, S k=1 ζ i (2, j, k). • Expected number of customers lost before entering the orbit per unit time, S k=1 ζ i (c, j, k). • Expected number of customers lost after retrials per unit time, S k=1 iζ i (c, j, k). • Expected number of priority generated customers lost per unit time,

Cost Analysis
In this section, we propose an optimization problem and explaining with a numerical example. Let us consider the cost and revenue variables as follows: • A revenue of r 1 monetary units for each ordinary customer getting service and leaving the system. • A revenue of r 2 monetary units for each priority generated customer getting service and leaving the system. Define a revenue(profit) function as : Our goal is to find an optimum (s, S) pair(with all other parameters fixed) that maximizes the expected total profit ET P. We provide a numerical example in the following section.

Numerical illustration
By variation of underlying parameters, we calculate the system characteristics numerically in this section.  Table 1 and 2. For another set of selected (s, S) entries with λ = 20.0 is tabulated in Table 3

Conclusion
An inventory system with special focus on self-generation of priorities studied in this paper. The arrival follows the Poisson process, and the single server provides service with different exponential distribution. Here we consider a retrial inventory and give a probability rule for returning to orbit after an unsuccessful retrial. Important system characteristics calculated for a suitable system designing and numerically illustrated with examples. An optimization problem is also discussed by introducing a profit function and calculated optimum (s, S) pairs.