Queues with Workload-Dependent Arrival and Service Rates

Abstract

We consider two types of queues with workload-dependent arrival rate and service speed. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.First, in the M/G/1 case, we compare the steady-state distribution of the workload (both at arbitrary epochs and at arrival instants) in two models, in which the ratio of arrival rate and service speed is equal. Applying level crossing arguments, we show that the steady-state distributions are proportional. Second, we consider a G/G/1-type queue with workload-dependent interarrival times and service speed. Using a stochastic mean value approach, several well-known relations for the workload at various epochs in the ordinary G/G/1 queue are generalized.

On/Off Storage Systems With State-Dependent Input, Output, and Switching Rates

“…Scheinhardt et al [14] allow a finite number N ≥ 2 of background states, but restrict themselves to a finite buffer content. The paper is also related to [4]. When restricting consideration of our model to the states in which the buffer content is decreasing, one obtains a queue with workload-dependent arrival and service rates, and with service requests that depend on the workload found at the time of their arrival.…”

“…When restricting consideration of our model to the states in which the buffer content is decreasing, one obtains a queue with workload-dependent arrival and service rates, and with service requests that depend on the workload found at the time of their arrival. In [4] queues with workload-dependent arrival and service rates have been studied, mainly with the restriction of i.i.d. service requests.…”

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

“…• We extend the problem addressed in [4,6] to a more general setting with workload-dependent MAP arrivals as opposed to Poisson arrivals.…”

“…In related work, the authors of [4] study an M/G/1 queue with workload-dependent arrivals and service rates for the infinite queue capacity case. The workload-bounded M/G/1 buffer under a complete rejection policy was studied in [5] with closed-form expressions for the M/M/1 case.…”

“…In this case, U is said to possess a PH-type distribution, which is characterized by the matrix pair (α, T ). As in [4], we assume that the service speed is represented by a function c(x) > 0 when the current workload takes the value x > 0, and c(0) = 0. When c(x) = c, then the service time has a phase-type distribution with characterizing pair (α, cT ), but otherwise there is no direct relationship between the job size and the service time.…”