| Along with the thorough research on queue theory and the complication of applicatio- n, the queue model describing queue phenomenon is becoming increasingly complex, and the rules introduced are also increasing, developed from single arrival and single service to bulk arrival and bulk service. The queue model with bulk arrival and bulk service is used in many aspects, such as the transport of goods in railway, the transport of goods in port and so on.At present, on the research of batch systems, people often combine bulk arrival or bulk service with many rules, such as vacation, repairable queue, a variety of control strategies and so on. The article [1] has studied the Mξ/M/m model with bulk arrival, in which one service window only serves one customer every time. The article [2] has studied the M/Mr/m model with bulk service, in which the number of customers served by one service window every time is r.On the basis of the articles [1],[2], this article discusses the Mξ/Mr/m queue. In the model, coustomers arrive in batches of random sizeξ, and r ,the number of customers served by one service window in one time, is a positive integer. In the paper, namely, X(t) is the number of coustomers in the system at time t. Firstly, the paper proves that {X(t),t≥0} is the birth-death process, gets its transfer probability matrix, and then solves that the condition in which the stationary distribution of the system exists isρ<1 by analyzing the state transition diagram. In the condition of steady state, further we give the smooth equations and solve them, at last we get the probability generating function of the system size and relevant indicators, for example, the average size of the system, the averaged dwell-time. Finally, letξ, r, m take special values, we find the results are same with the existing model. To some degree, that verifies the results of this paper are correct. So we can say this paper promotes the related results of the articles [1],[2]. |
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