SOME LARGE DEVIATION RESULTS IN REAL LIFE WITH SPECIAL REFERENCE TO QUEUES AND QUEUEING-INVENTORY

We consider an $G/M/1$ - type queue where inter-arrival times have general distribution, depending on the type of arrival. Three types of customers arrive to the system. The first type is genuine customer, second mild destructive (negative customer) which throws out the customer in service but adds nothing and the third one is the most destructive which wipes out all customers from the system. $p_i$ is the probability that the type $i$, $(i = 1, 2, 3)$ customer arrives. The distribution of the inter-occurrence time of arrivals is generally distributed with $G(.)$ as the distribution function. Thus $G_i(t)$ is the probability that an arrival takes place and it is of type i where $G_i(t) = p_i G(t)$. Service time of genuine customers follows exponential distribution. This system is analysed as a G/M1- type queue which can easily be seen as stable.

Next, we introduce correlated arrival process in the above. Arrival process is MMAP with representation $(D_0, D-1, D_2, D_3)$. $D_0$ contains rates of transitions without arrival; its diagonal entries are non-positive and off-diagonal entries, non-negative; $D_1$ has transitions rates of arriving genuine customers; $D_2$, that for negative customers which is not highly destructive and $D_3$, that of the most destructive customers. Here also $p_i$ stands for the probability of an arrival of type $i$. This system is always stable. This system is analysed to find the long run distribution of the system state. Performance characteristics of the system are also investigated.