Single Server, Multi-Class Queues with Markovian Arrival and Ambiguity of Class Determination

Abstract

In this paper we introduce a queueing system manned by a single server who serves multiple class(n)of customers. The customers arrive according to a Markovian arrival process and form a single class(n)of customers. The customers arrive according to a Markovian arrival process and form a single queue. At the time when taken for service the customer of class i may be taken for service of class j queue. At the time when taken for service the customer of class i may be taken for service of class j (ambiguity in the determination of class of required service) with probability p , 1 ≤ j ≤ n . Service time (ambiguity in the determination of class of required service) with probability pij , 1 ≤ j ≤ n . Service time (i) (i) (i) in classi is of phase type distributed with representation(γ ,T )of order m ij , 1≤ i ≤ n. If a customer in class i is of phase type distributed with representation (γ i , Ti ) of order mi iii, 1 ≤ i ≤ n . If a customer of class i is taken for class j service initially then on completion of service there he is taken to class of class i is taken for class j service initially then on completion of service there he is taken to class (i) with probability η(i),1≤k≤n. A timer starts at the beginning of service of a customer. If the timer with probability ηjk , 1≤ k ≤ n. A timer starts at the beginning of service of a customer. If the timer realizes before the customer is identified of the required (correct) service, this customer is instantly sent realizes before the customer is identified of the required (correct) service, this customer is instantly sent out of the system without getting correct service. On the other hand if the timer does not realize before out of the system without getting correct service. On the other hand if the timer does not realize before identification of required service, then the customer is taken for service in that class and completes identification of required service, then the customer is taken for service in that class and completes service successfully. In the case when the customer is provided the required class of service right from service successfully. In the case when the customer is provided the required class of service right from the very beginning, he leaves the system after completing this service. In the last case timer plays no the very beginning, he leaves the system after completing this service. In the last case timer plays no role in the service time of such customers. In the case of such customers no ambiguity arises on the type role in the service time of such customers. In the case of such customers no ambiguity arises on the type (class) of service required. We analyze the above system to derive the expected time a customer spends (class) of service required. We analyze the above system to derive the expected time a customer spends with the server. Then we use it to derive the stability condition and the resulting system state with the server. Then we use it to derive the stability condition and the resulting system state distribution. Useful performance indices are computed. Numerical illustrations are provided to have a distribution. Useful performance indices are computed. Numerical illustrations are provided to have a glimpse of the system performances. Some examples from real life situation are cited, as motivation for glimpse of the system performances. Some examples from real life situation are cited, as motivation for the study of the above mentioned model. Case of arbitrarily distributed service time is also considered. the study of the above mentioned model. Case of arbitrarily distributed service time is also considered. An application in telecommunication is indicated. An application in telecommunication is indicated.