Analysis of Inventory Models

Abstract

The thesis is divided into 6 chapters. The first chapter is Introduction. In this chapter the basic concepts of queuing models and inventory queuing models are explained. In the second chapter we give a brief outline of the related works in the area. The third chapter is titled “An Explicit Solution for an Inventory Model with Positive Lead Time and Backlogs”. Here we consider an (s, S) inventory model in which customers arrive to a single server counter according to a Poisson process where inventory served. Inventory is replenished according to (s, S) policy, the replenishment time being an exponential random variable. We assume negligible service time for this model. The customers who join the queue when inventory level drops to zero form a queue and remain in the system until inventory replenishment is realized. The explicit expression for the steady state probability vector has been derived. The expression for expected waiting time of a customer in the queue has been derived.A numerical study of the effect of parameters on the performance measures has been done. The fourth chapter is titled “An Explicit Solution for an Inventory Model with Positive Lead Time and Server Interruptions”. In this model we consider a single server queuing system with inventory.Customers arrive according to a Poisson process and service times follow exponential distribution. Inventory is replenished according to (s, S) policy with positive lead time which follows exponential distribution. While the server serves a customer, the service may be interrupted; the interruption time follows an exponential distribution. Following an interruption the service restarts after repair at an exponential rate. We assume that while the server is on interruption, the customer being served waits there until his service is completed,no inventory is lost due to interruption, no arrivals are allowed when the server is on interruption and an order placed if any is cancelled.Stability of the above system is analyzed and the steady state probability vector is calculated explicitly. Expressions for several system performance measures such as expected number of customers in the system, expected inventory level, expected interruption rate etc. are obtained. Even though explicit expressions are obtained, a numerical study of the effect of parameters on the performance measures has been done. A cost analysis has also been done for the model. The fifth chapter is titled “An Explicit Solution for an Inventory Model with Server Interruption and Retrials”. In this model customers enter into a single server queuing model in accordance with a Poisson process where inventory is served. The inter service time follows exponential distribution. Upon arrival, finding the server busy the customers enter into an orbit from where they retry for service at a constant retrial rate. While the server serves a customer the service can be interrupted, the inter occurrence time of interruption being exponentially distributed. Following a service interruption the service restarts after an exponentially distributed time. Inventory is replenished according to (s, S) policy, replenishment being instantaneous. For the model under discussion we assume that no inventory is lost due to server interruption, the customer being served when interruption occurs waits there until his service is completed and no arrivals are entertained and an order placed if any is cancelled while the server is on interruption. Explicit expression for the steady state probabilities is calculated and several performance measures are evaluated explicitly and numerically. Graphs which show the variation of various performance measures with parameter values are also drawn. In the sixth chapter we provide some recommendations about the future work and extensions which can be done based on the models discussed in the thesis.