Structural analysis of a queueing system with multiclasses of correlated arrivals and blocking

In assemble-to-order production systems, and others of a similar flavor, it is often the case that orders for components of various types are placed simultaneously, but that these components are manufactured or assembled at separate facilities. The order process introduces correlation among the jobs at separate facilities. The purpose of this paper is to study the effect of this correlation on a variety of system performance measures. Consider a system that consists of s parallel servers, where each server has a finite buffer and is dedicated to a separate job type. Multiple classes of customer orders arrive to the system, where each class is composed of one or more unique job types. Upon the arrival of an order, each job in the order is separately routed to its designated buffer; if the buffer is full, that job is blocked and lost; otherwise, it enters the buffer and is served according to the FCFS discipline. Under Markovian assumptions, we systematically examine the impact of arrival correlations on system-based performance measures such as the queue length vector and the workload vector and class-based performance measures such as the waiting time vector and the order response time. Among other things, we establish several stochastic orders between performance vectors with different degrees of arrival correlations. We also show that greater arrival correlation can stochastically improve the worst component in a performance vector (e.g., the longest queue, the heaviest workload), reduce the expected sum of the j longest queues, 1 less than or equal to j less than or equal to s, and, for any given order type, increase its entering probability and reduce its order response time. Our results can also be extended to the compound Poisson arrival process, where each order contains multiple units of several job types.