TY - JOUR

T1 - Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

AU - Nazarathy, J.

AU - Weiss, G.

PY - 2010

Y1 - 2010

N2 - We consider a push pull queueing network with two servers and two types of job which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar–Seidman Rybko–Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of job are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with an infinite supply of work.
Keywords: Queueing networks; Push pull; Infinite virtual queues; Fluid models; Positive Harris recurrence; Diffusion limits; Petite bounded sets.

AB - We consider a push pull queueing network with two servers and two types of job which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar–Seidman Rybko–Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of job are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with an infinite supply of work.
Keywords: Queueing networks; Push pull; Infinite virtual queues; Fluid models; Positive Harris recurrence; Diffusion limits; Petite bounded sets.

U2 - 10.1016/j.peva.2009.09.010

DO - 10.1016/j.peva.2009.09.010

M3 - Article

VL - 67

SP - 201

EP - 217

JO - Performance Evaluation

JF - Performance Evaluation

SN - 0166-5316

IS - 4

ER -