Approximating a Point Process by a Renewal Process, II: Superposition Arrival Processes to Queues

Abstract

We develop an approximation for a queue having an arrival process that is the superposition of independent renewal processes, i.e., ∑GI1/G/1. This model is useful, for example, in analyzing networks of queues where the arrival process to an individual queue is the superposition of departure processes from other queues. If component arrival processes are approximated by renewal processes, the ∑GI1/G/1 model applies. The approximation proposed is a hybrid that combines two basic methods described by Whitt. All these methods approximate the complex superposition process by a renewal process and yield a GI/G/1 queue that can be solved analytically or approximately. In the hybrid method, the moments of the intervals in the approximating renewal process are a convex combination of the moments determined by the basic methods. The weight in the convex combination is identified using the asymptotic properties of the basic methods together with simulation. When compared to simulation estimates, the error in hybrid approximations of the expected number in the queue is 3%; in contrast, the errors of the basic methods are 20–30%. The quality of the approximations suggests that the hybrid approach would be useful in approximating point processes in other contexts.

Approximating Multiple Arrival Streams by Using Aggregation

“…(iii) the GIIGII system where the first two moments of the inter-arrival time are determined according to the method of Whitt [11] and Albin [1].…”

“…Now we can not compare it with the system with a renewal demand process, the first two moments of the inter-arrival times of which are determined by the method of Whitt [11] and Albin [1]. The reason is that their method is specifically designed for 'EGi/Gll queues.…”

“…In general, this approximation can lead to severe errors. A more sophisticated method has been developed by Albin [1] and Whitt [11]. They also approximate the superposition by a renewal process, but the second moment of the inter-arrival time is determined differently: the squared coefficient of variation of the inter-arrival time is determined as a convex combination of the squared coefficients of variation obtained from the so-called asymptotic and stationary-interval approximations.…”

Optimal Decentralization of Early Infant Diagnosis of HIV in Resource-Limited Settings

“…I approaches exponential distribution when the number of superposing processes is large (Albin 1982(Albin , 1984 and assess the quality of this approximation. Next, we provide an exact characterization of X under the assumptions made in this section.…”

Modeling a Hospital Queueing Network

“…It was pointed out by Albin (1984) that if at least one of the interarrival time distributions, constituting the arrival process, does not stem from a Poisson process, the resulting aggregate interarrival times do no longer hold the property of independence. As a result the analytical analysis of the aggregate arrival process becomes highly intractable.…”