Queueing Systems With Negative Customers In Heavy-traffic

In this paper we establish the heavy-traffic limits for queueing systems with negative customers, including single-server queues and multiserver queues. For the single-server queue system with negative customers, we obtain the heavy-traffic FCLTs both for the standard single-server queues (arrival from single source) and queues with superposition arrivals (arrivals from multiple sources), by applying the continuous-mapping approach to establish proper heavy-traffic stochastic-process limits for the queue-length processes in queueing models. Meanwhile, we show two special cases (the Brownian case and the stable-Levy-motion case) of the heavy-traffic limit theorem and obtain the results of the reflected-Brownian-motion limit and the reflected-stable-Levy-motion limit respectively. In addition, we will use matlab to simulate and plot the queue-length processes of models such as the M/M/1 queue with negative customers, the M/P1.5/1/ queue with negative customers and the P1.5/M/1 queue with negative customers. Thus the conclusions can be verified. We also compare the computer simulations of the mean queue-length to the exact values and the approximations in the M/M/1 queue model with negative customers. The results of our numerical comparisons show that the simulations are more close to the exact values and the approximation performs badly whenρn is relatively low, whileρn becomes close to 1, the simulations become close to the approximations, so that the approximation performs spectacularly well. For the multiserver queue system with negative customers, there are two principal cases:first, when there is a moderate number of servers and, second, when there is a large number of servers (infinite number). We consider the first case of a fixed finite number of servers by introducing the autonomous service models. Because the reflection map can be applied directly, we can obtain the heavy-traffic limit for the s-server queue with negative customers and autonomous service firstly, then we show that the queue-length processes in the standard queue and the queue with autonomous service are asymptotically equaivalent. Moreover, we can find that the heavy-traffic behavior of this fixed finite number of severs is essentially the same as a single-sever system. We consider the second case of a large number of servers by suppose that the service times take values in a finite set, so that we can analyze the queue-length process in terms of the arrival counting process and then obtain the heavy-traffic limit.