`Journal of Applied Mathematics and Stochastic AnalysisVolume 2007, Article ID 37848, 23 pageshttp://dx.doi.org/10.1155/2007/37848`
Research Article

## Comparison of Inventory Systems with Service, Positive Lead-Time, Loss, and Retrial of Customers

1Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala 682-022, India
2Department of Mathematics, Saint Peter's College, Mahatma Gandhi University, Kolenchery, Kerala 682-311, India

Received 22 November 2006; Accepted 21 June 2007

Copyright © 2007 A. Krishnamoorthy and K. P. Jose. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.