Queueing-Inventory with One Essential and m Optional Items with Environment Change Process Forming Correlated Renewal Process (MEP)

We consider a queueing inventory with one essential and m optional items for sale. The system evolves in environments that change randomly. There are n environments that appear in a random fashion governed by a Marked Markovian Environment change process. Customers demand the main item plus none, one, or more of the optional items, but were restricted to at most one unit of each optional item. Service time of the main item is phase type distributed and that of optional items have exponential distributions with parameters that depend on the type of the item, as well as the environment under consideration. If the essential item is not available, service will not be provided. The lead times of optional and main items have exponential distributions having parameters that depend on the type of the item. The condition for stability of the system is analyzed by considering a multi-dimensional continuous time Markov chain that represent the evolution of the system. Under this condition, various performance characteristics of the system are derived. In terms of these, a cost function is constructed and optimal control policies of the different types of commodities are investigated. Numerical results are provided to give a glimpse of the system performance.


Introduction
Queueing inventory models have been extensively analyzed since 1992. Very few of these discuss multi-commodity systems in randomly changing environments. Queueing systems which evolve under influences from external sources have, for a long time, inspired interest. In real life situations, inventory systems are often subject to randomly changing exogenous environment conditions that affect the demand for the product, the supply, and the cost structure. The area of queues in random environments is today a field of active research in applied probability. Queueing systems with correlated arrival flow of customers give adequate mathematical models for different real world systems including computer and telecommunication systems, and network protocols [1]. The following papers are relevant to the present paper only in that the authors consider multi-commodity inventory systems without any specified main and optional items. FaizAl-Khayyal et al. [2] consider a multi-commodity network model in maritime routing and scheduling. They tried to The following papers deal with queueing inventory systems influenced by randomly changing environments. In Song et al. [9], the authors consider an inventory model where the rate of demand is dependent on the environment variables. These variables can be anything, such as different stages in the life cycle of the particular inventory or changes in various factors linked with the economy, etc. They not only derived basic characteristics of the optimal policies but also observed the influence of various patterns in problem data on optimal policies and developed algorithms for computing optimal policies Özekici et al. [10] describe inventory models with unreliable suppliers in randomly changing environments. The environment change follows a Markov chain. The dependence of the stock-flow equations of the system on random environments is represented by a two-dimensional stochastic process. Under specified conditions, they have derived an optimality condition for the base-stock policy and (s, S) policy. Computational issues and some extensions are also determined.
A single item inventory model which is observed periodically in a randomly changing environment is considered in Erdem et al. [11]. All the model parameters are dependent on a time-homogenous Markov chain environment. The replenishment quantity is min-imum{Order quantity, Vendors capacity}. The problem is analyzed in single, multiple, and infinite periods. In all these cases, the authors prove that the optimal base-stock level depends on the state of the environment. Comparisons of the results with the case when the replenishment quantity equals the quantity ordered is also done.
Perry et al. [12] discuss production-inventory models with an unreliable facility operating in a two-state random environment. The system is characterized by a production machine. The production can even be stopped purposefully when there is a limited stocking capacity. When the machine is in ON period, the input into the buffer is assumed to be continuous and uniform until the threshold is reached, whereas the output from the buffer follows a compound Poisson process during OFF periods. Two different models are discussed and the factors controlling OFF periods are determined.
A continuous review (s, S) inventory system in a randomly changing environment is discussed in Feldman et al. [13] and its steady-state distribution obtained. The demand process is an environment dependent compound Poisson process when the environment is in a fixed state during an interval of time. The environmental process follows a continuoustime Markov process.
Kalpakam et al. [14] consider a lost sales (s, S) inventory system in a random environment. No backlog is allowed. The demand and supply rates are influenced by the environment process which is a finite irreducible Markov chain in continuous time. They have obtained the transform solution of the inventory level distribution and also an efficient algorithm to evaluate the long run system state is provided. Moreover, transient and limiting values of the mean reorder and shortage rates are also obtained. Goh et al. [15] discuss price-dependent inventory models with discount offers at random times. The offer is accepted when the inventory position is lower than a threshold level. Three different pricing policies are considered in which demand is induced by the retailer's price variation. They have obtained expressions for optimal order quantities, prices, and profits under the assumptions of constant demand rates.
Highlights of this paper are: • It considers multi-commodity inventory with positive service time [16] in finite number of randomly changing environments; • The first paper to introduce optional items for service in random environments; • Except for one item (essential), all others are optional; • The customer demand process follows Markovian arrival process (MAP); • The environment change process follows marked Markovian environment arrival process (MMEAP)[n] of order n; • Service time of customers, being served with the essential inventory follows phase type distribution and that w.r.t optional item(s) follows exponential distribution (depending on the environment). The latter has a parameter, depending on the specific item(s) demanded by the customer.
The rest of the paper is organized as follows. The mathematical formulation of the model including the stability condition and the steady-state probability vector is described in detail in Section 2. Section 3 deals with some system performance measures and in Section 4, the construction of the cost function for optimizing the system control variables is discussed. Numerical illustrations and the numerical analysis of the cost function are discussed in Section 5. Section 6 gives the conclusion followed by references.
Notations and abbreviations used: • (s, S) ordering policy: An inventory policy which says that when the inventory level falls below a certain minimum number s, the order for replenishment is made to restore the inventory to a maximum number S; • e = Column vector of appropriate order with all its entries as 1 s; •0 = Matrix of appropriate order with all its entries as 0; • I k = Identity matrix of order k; • [G] pq = (p, q)th element of the matrix G;

Mathematical Formulation
Consider a single server multi-commodity queueing inventory system with one essential and m optional inventories in n random environments. Only one environment will be in operation at any given time. The arrival of customers follows Markovian arrival process (MAP) with representation (H 0 , H 1 ), where each H i for 0 ≤ i ≤ 1 is of order m 3 . The generator matrix of the underlying CTMC (δ(t), t ≥ 0) on the state space {1, 2, 3, . . . , m 3 } is given by H = H 0 + H 1 . These matrices, H 0 and H 1 , are of the form ij , 1 ≤ i, j ≤ m 3 gives the transition rate from ith state to jth state through an arrival, while h (0) ij , 1 ≤ i, j ≤ m 3 , gives the transition from ith state to jth state without an arrival. Note that the transition rate between the ith states, given by h (1) ii , 1 ≤ i ≤ m 3 occurs only with an arrival. Let η be the steady-state probability vector of H. Then, η satisfy ηH = 0 and ηe = 1. The fundamental rate λ A of this MAP is given by λ A = ηH 1 e which gives the expected number of arrivals per unit of time. The coefficient of variation C var of intervals between arrivals is calculated as C var = 2λ A η(−H 0 ) −1 e − 1 and coefficient of correlation C cor of intervals between successive arrivals is given as There are n environments that occur randomly and the occurrence of the environments follows marked Markovian environment arrival process (MMEAP[n]) with representation (D 0 ,D 1 ,D 2 ,. . . ,D n ), where each D i for 0 ≤ i ≤ n is of order m 2 . As stated earlier, only one environment will be in operation at any given time. The change in environment is directed by the stochastic process {V (t); t ≥ 0} which is an irreducible continuous time Markov chain with the state space {1, 2, . . . , m 2 }. The sojourn time of this chain in the state v, 1 ≤ v ≤ m 2 , is exponentially distributed with parameter λ (v) . When the sojourn time in the state v expires,the process {V (t); t ≥ 0} jumps to the state v without any change in the environment with probability p On the other hand, the process {V (t); t ≥ 0} jumps to the state v with the arrival of lth environment with probability p The behavior of the MMEAP is completely characterized by the matrices D l , l = 0, 1, 2, . . . n defined by . . , n} The matrix D = ∑ n l=0 D l represents the generator of the process {V (t); t ≥ 0}.
Service time of those customers who are served with the essential item is phase type distributed with representation PH(γ, T) of order m 1 . This service time is the time until the undergoing Markov chain (ζ(t), t ≥ 0) with a finite state space {1, 2, 3, . . . , m 1 + 1} reaches the absorbing state m 1 + 1. γ = (γ 1 , γ 2 , . . . , γ m 1 ) gives the initial probability of starting in any of the m 1 states. T is the generator matrix that gives transition rates within the states {1, 2, 3, . . . , m 1 }. The absorption rates from the individual transient states {1, 2, 3, . . . , m 1 } to the absorption state m 1 + 1 is given by T 0 = −Te. µ = −γT −1 e gives the mean service of the customer.
Service time of those customers who are served with the optional items are environment dependent and they are exponentially distributed with parameter µ k i , where 1 ≤ k ≤ n and i ∈ {i 1 , (i 1 i 2 ), (i 1 i 2 i 3 ), . . . , (i 1 i 2 i 3 . . . i m )}. It is important to note that no order preference has been given to any element, i.e., (i j i l ) = (i l i j ), and so on where each i l ∈ {1, 2, 3, . . . , m} with 1 ≤ j = l ≤ m.
In this model, a customer is allowed to demand exactly one unit of the essential inventory where as more than one type of optional inventory can be demanded by a customer with an imposed restriction of, at most, one item from each optional inventories. Service rates of the optional items are assumed to be environment dependent. The ith optional item is served in the kth environment with probability p k i , similarly the lth and rth optional inventories are served in the kth environment with probability p k lr , and so on. If the demanded optional inventory is not available, the customer is expected to quit the system after acquiring the essential item together with those available optional inventories. The server is assumed to be in the idle state in the absence of customers, as well as essential inventories. Essential and optional inventories have exponentially distributed positive lead time with parameters β and β j for 1 ≤ j ≤ m, respectively. The essential inventories are under the (s, S) control policy whereas the environment dependent optional inventories are under the (s k i ,S i ) for i = 1, 2, 3, . . . , m and k = 1, 2, 3, . . . , n control policies in the kth environment.
At any given time t, let N(t), S(t),E(t), O l (t),J 1 (t),J 2 (t), and J 3 (t) denote, respectively, number of customers in the system, status of environment, number of essential inventory items, number of lth optional inventory items for 1 ≤ l ≤ m, service phase, environment phase and arrival phase of the customers. The status of the server at any given time t is defined as, Essential inventory service l lth optional inventory service . . . Let ∆ be the collection of all the permitted combinations of different optional inventories and let C u denotes the server status, in general, for the combined service of u optional items, for 1 ≤ u ≤ m. Thus, the process The infinitesimal generator Q of the system is of the form Matrices A 01 and A 0 are of order a × b and a × a, respectively, their entries are due to the arrival of customers following MAP with representation (H 0 , H 1 ). Matrices A 10 and A 2 are of order c × a and c × c, respectively, their entries are due to the service of essential inventories following phase type distribution with representation PH(γ, T) and also due to the environment dependent, exponentially distributed service of optional inventories. Matrices A 00 and A 1 are square matrices of order a and c, respectively, their entries includes the replenishment rates of the inventories in addition to the negative sign of sum of other entries of the same row found in A 01 , A 0 , A 10 , and When u = 12 . . . m, then l 12...m = π m k=1 S k m 2 m 3 . In order to have a better understanding of the system, a detailed illustration of the model has been provided in Appendix A by fixing the number of optional items m = 2 and the number of environments n = 2. All the transitions and resultant component matrices are shown clearly in the Appendix A.

Stability Condition
Let π = (π 0 , π 1 , π 2 , . . . , π S ) be the steady-state probability vector of Refer to Appendix A for the component matrix representations From (1), where, The only unknown probability vector π S is obtained from the normalizing condition Theorem 2.1. The necessary and sufficient condition for the stability of queuing inventory system under study is Proof. The queueing system with the generator Q under study is stable if, and only if, Refer to Appendix A for the component matrix representations.
Using Equations (6)-(9) together with the matrices A 0 and A 2 we get Then, by (10) we get the stated result.

Steady State Probability Vector
Let x denote the steady state probability vector of the generator Q. Then, we have Partitioning x as x = (x 0 , x 1 , x 2 , ...), from (23) we get By assuming the stability condition, we see that x is obtained as (see [17]) where R is the minimal non-negative solution of the matrix quadratic equation The boundary conditions are given by From Equation (23) we get, and by the normalizing condition in (23), we get

1.
Expected re-ordering rate of the essential inventory Expected re-ordering rate of lth optional item in the ath environment, Expected number of essential inventories in the system

5.
Expected number of lth optional inventories in the system for 1 ≤ l ≤ m.
Expected loss rate of customers in the absence of essential item

Cost Function
In-order to optimize the inventory levels s, S, s k i , and S i for 1 ≤ i ≤ m, 1 ≤ k ≤ n, we construct the following cost function, C EI = Carrying cost per unit of the essential item; 4.
C OI (i) = Carrying cost per unit of the ith optional item; 5.
C 1 = Customer holding cost per unit time; 6.
C 2 = Cost due to loss of goodwill per unit time, in the absence of the essential item.

Numerical Illustration
In this section, we provide the numerical illustration of the system performance measures with varied values of the underlying parameters. The model with one essential and two optional inventories is considered here.

Conclusions
We studied a single server multi-commodity queueing inventory system with one essential and m optional items in n random environments. The condition for stability of the system is obtained. Under this condition, different performance measures of the system are derived. A cost function involving these measures and inventory control variables is constructed. Optimization of the cost function along with the control variables is also done numerically. The obtained numerical results showed huge resemblances with what we see and experience around us. A very familiar example is a car/truck (Heavy automobiles) showroom, where one can see only very few items (main item) displayed. In this example, one can think of the additional accessories as the optional inventories. Here, booking of the essential inventory with other optional inventories has to be done as per the requirement of the customer.

2.
Transition rates due to the service completion of essential and optional items.
(a) Transition rates from level 1 to level 0 due to the service completion of optional inventory i.

Submatrices of A ij 00
For i = j, Submatrices of A ij 01 For i = j,Ĝ 2 = G 2 .