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 (1−γ). 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 (1−δ). 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 (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). 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.
Highlights
In all works reported in inventory prior to 1993, it was assumed that the time required to serve the items to the customer is negligible
In addition to the description in model I, we provide a buffer of varying capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III
Krishnamoorthy and Islam [11] analyzed a production inventory with retrial of customers. They considered an (s, S) inventory system where arrivals of customers form a Poisson process and demands arising from the orbital customers are exponentially distributed with linear rate
Summary
In all works reported in inventory prior to 1993, it was assumed that the time required to serve the items to the customer is negligible. In model I, an arriving customer who finds the inventory level zero or server busy proceeds to an orbit with probability γ and is lost forever with probability (1 − γ). A retrial customer in the orbit who finds the inventory level zero or server busy returns to the orbit with probability δ and is lost forever with probability (1 − δ). A retrial customer from the orbit who finds buffer full returns to the orbit with probability δ and is lost forever with probability (1 − δ) In all these cases, the interretrial times follow an exponential distribution with linear rate iθ when there are i customers in the system.
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