Generalized Precedence-Based Queueing Systems

Abstract

A general K-server queueing model with precedence constraints on customer services is considered. Such a system is characterized by bulk arrivals of size K, with one customer for each server, and synchronization constraints between them and some of the previously arrived customers: Customer k (1 ≤ k ≤ K) of the nth (n ≥ 0) bulk arrival has to wait for the service completion of customer j (1 ≤ j ≤ K) of the mth (0 ≤ m ≤ n) arrival before starting its service, if there exists a precedence relation between these two customers. Such queueing systems arise naturally when modeling database concurrency control mechanisms based on “static” locking, as well as parallel computations where some processes may need results obtained by other processes for beginning their execution. Several performance aspects of this class of queueing systems are considered under various statistical assumptions. The stability condition, which yields the maximal system throughput, is analyzed together with the waiting and response times of individual customers, and the response times of bulks. These results are obtained from a set of recursive evolution equations which captures both the queueing and the synchronization mechanisms of the system. The stability condition is derived from a generalization of the schema of Loynes for G/G/1 queues. Integral functional equations are provided for both the transient and stationary distributions of the waiting and response times under renewal type assumptions. Various upper and lower bounds on the moments of these quantities are derived using either modifications of the precedence function or stochastic ordering techniques.