Research On Random Service Systems Of The Batch Arrival

Classical random service system (queueing system), originated from the research of the telephone communication using the mathematic theory by Danish mathematician Erlang at the beginning of the 20th century.The theory of random service system has acquired enormous development in past years, and became one of the most vigorous research fields in the applying probability and the random operations research. Not only does it have a comparatively complete theoretical system, but also has extensive application in such each field as military, economy, production, management, traffic, etc. Classical random service system, including the forms such as M/M/l,M/G/l,GI/G/l,GI/M/n, has been made deep research by many experts whose majors are the applying probability and the random operations and got many perfect results.In recent years, a lot of scholars began to extend the classical models to the other research fields through various ways and studied some more complicated models on the basis of classical queueing theories. These amplification includes the following several respect mainly: Firstly, it is considered that the general assumption of the customers' arrival and the service time of the queue models. For example, some scholars suppose the arrival process of the customers as Markov renew process or PH process; Secondly, some scholars consider the queueing system with the priority classes of customers. At last, the vacation and repairable system is also considered as a part of complicated queueing system.There is a important hypothesis in the research of classical queueing system: the arrival process of the customers can be described by the random point processes which haven't more than one arrival in one point. But the hypothesis maybe be untenable in many situations ,for example, the batch arrival of the information flow in the communication technology; the batch inspection of the product in some factories.This paper is based on the facts introduced above. On the basis of the classical queueing system, we consider the batch arrival of the customers in the system and get some conclusion. At first, we consider the batch arrivals of the customers submit to a discrete distribution , and get the queue length distribution of the model. At the same time, the waiting time and the busy time of the system is considered.Secondly, it is considered that the batch arrivals of the customers submit to a continue distribution in the queueing system. The batch arrivals depend on the Poisson distribution, the customers arrive in one batch are random variables and the service times have a common continue distribution. Using the method of the embedded Markov Chain, It is given that the distributions of the queue length and the waiting time in the system. At the same time, the distribution of the busy time is obtained.At last, the paper considers the continuous queue with nonpreemptive priority. The arrival processes of the two priority classes are the batch arrival according to the Poisson processes.The service times of customers in each priority class are i.i.d. according to a general probability distribution which may differ between the two priority classes. Using the supplementary variable method and the generating function technique, we derive the joint probability generating function of the stationary waiting time distribution of the customers.