Analysis of Some Stochastic Inventory System in Supply Chain

In this chapter we consider a three echelon inventory control system which is modeled as a warehouse, single distribute and one retailer system handling a single product. A finished product is supplied from warehouse to distribution center which adopts one-for-one replenishment policy. The replenishment of items in terms of packets from warehouse to distribution center with exponential lead time having parameter µ1. Then the product is supplied from distribution center to retailer who adopts (s, S) policy. Supply to the retailer in packets of Q (= S - s) items is administrated with exponential lead time having parameter µ0. The demand at retailer node follows a Poisson with mean lambda. The steady state probability distribution of system states and the measures of system performance in the steady state are obtained explicitly. The Cost function is computed by using numerical searching algorithms, the optimal reorder points are obtained for various input parameters. Sensitivity analysis are discussed for various cost parameter such as holding cost, setup cost etc.

Analysis of Feedback Retrial Queue with Starting Failure and Server Vacation

In this chapter we discusses a batch arrival feedback retrial queue with Bernoulli vacation, where the server is subjected to starting failure. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters a service immediately while the rest join the orbit. After the completion of each service, the server either goes for a vacation with probability or may wait for serving the next customer. Repair times, service times and vacation times are assumed to be arbitrarily distributed. The time dependent probability generating functions have been obtained in terms of their Laplace transforms. The steady state analysis and key performance measures of the system are also studied. Finally, some numerical illustrations are presented.

A Single Server Retrial Queueing System with Two Types of Batch Arrivals

A retrial queueing system with two types of batch arrivals is considered. The arrivals are called type I and type II customers. The type I customers arrive in batches of size k with probability c_k and type II customers arrive in batches of size k with probability d_k. Service time distributions are identical independent distributions and are different for both type of customers. If the arriving customers are blocked due to server being busy, type I customers are queued in a priority queue of infinity capacity whereas type II customers entered into retrial group in order to seek service again after a random amount of time. For this model the joint distribution of the number of customers in the priority queue and in the retrial group in closed form is obtained. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

Performance Analysis of an M/G/1 Feedback Retrial Queue with Two Types of Service and Bernoulli Vacation

This chapter is concerned with the analysis of a single server retrial queue with two types of service, Bernoulli vacation and feedback. The server provides two types of service i.e., type 1 service with probability??1 and type 2 service with probability ??2. We assume that the arriving customer who finds the server busy upon arrival leaves the service area and are queued in the orbit in accordance with an FCFS discipline and repeats its request for service after some random time. After completion of type 1 or type 2 service the unsatisfied customer can feedback and joins the tail of the retrial queue with probability f or else may depart from the system with probability 1–f. Further the server takes vacation under Bernoulli schedule mechanism, i.e., after each service completion the server takes a vacation with probability q or with probability p waits to serve the next customer. For this queueing model, the steady state distributions of the server state and the number of customers in the orbit are obtained using supplementary variable technique. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.

Utility Maximization and Optimal Portfolio Selection

Utility maximization and optimal portfolio selection with or with-out consumption/transaction cost based on stochastic models of prices of securities with stochastic volatility are discussed.

On Sequential Decision Problems with Constant Costs of Observation

We present a solution technique for optimal stopping problems with constant costs of observation in a diffusion setting. Such problems arise naturally, e.g., in Wald's type sequential decision problems and the Portfolio optimization model by Morton and Pliska. The main result is that the treatment of such problem boils down to the determination of the maximum points of a class of explicitly given functions. The findings are illustrated by a variety of examples and generalized to random costs of observation.

Perishable Inventory System with Bi-Level Service System

In this chapter we consider a perishable inventory system under continuous review at a bi-level service system with finite waiting hall of size N. The maximum storage capacity of the inventory is S units. We assumed that a demand for the commodity is of unit size. The arrival time points of customers form a Poisson process. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The effect of the two modes of operations on the system performance measures is also discussed. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The items are perishable in nature and the life time of each item is assumed to be exponentially distributed. The demands that occur during stock out periods are lost. The joint probability distribution of the number of customers is obtained in the steady-state case. Various system performance measures in the steady state are derived. The results are illustrated numerically.

Stochastic Inventory System with Compliment Item and Retrial Customers

In this chapter, the author consider a two commodity stochastic inventory system under continuous review with maximum capacity of i-th commodity is Si(i=1,2). In this two commodity one is main item and the other is compliment item. It is assumed that demand for the i-th commodity is of unit size and demand time points form a Poisson process. The compliment item is supplied as a gift whenever the demand occurs for the main item, but no main item is provided as a gift for demanding a compliment item. Reordering for supply is initiated as soon as the on-hand inventory level of the main item reaches a certain level s1, and there is a lead time until the reorder arrives but instantaneous replenishment for the compliment item. The arriving any primary demands enter into an orbit, when the inventory level of main item is zero. The limiting probability distribution for both commodities and the number of demands in the orbit, is computed and various operational characteristics are derived. The results are illustrated with numerical examples.

M/Ck/1 Queue with Impatient Customers

A single-server queuing system with impatient customers and Coxian service is examined. It is assumed that arrivals are Poisson with a constant rate. When the server is busy upon an arrival, customer joins the queue and there is an infinite capacity of the queue. Since the variance of the service time is relatively high, the service time distribution is characterized by k-phase Cox distribution. Due to the high variability of service times and since some of the services take extremely long time, customers not only in the queue, but also in the service may become impatient. Each customer, upon arrival, activates an individual timer and starts his patience time. The patience time for each customer is a random variable which has exponential distribution. If the service does not completed before the customer's time expires, the customer abandons the queue never to return. The model is expressed as birth-and-death process and the balance equations are provided.

Substitutable Inventory System with Partial Backlogging under Continuous Review

Consider a two-commodity substitutable inventory system with storage capacity Si for commodity i, (i=1,2) under continuous review. The demand time points for each commodity are assumed to form independent Poisson processes. The two commodities are assumed to be substitutable. That is when any one of the commodity's inventory level reaches zero, then the demand for that commodity will be satisfied by the other commodity. If no substitute is available, then this demand is backlogged up to the level Ni, for commodity i, (i=1,2). The reordering policy is to place an order for both the commodities, when both inventory levels are less than or equal to their respective reorder levels. If the inventory level drops to N1 or N2, then both inventory levels are pulled back to their maximum levels S1 and S2 immediately and the previous order gets canceled. The lead time is assumed to follow negative exponential distribution. Various stationary measures of system performances have been derived and total expected cost rate is computed. Numerical examples are provided.