A Tandem Queue with Server Slow-Down and Blocking

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsCWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. A tandem queue with server slow-down and blocking ABSTRACT We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a blocking threshold . In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a slow-down threshold, which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{rho_ 1, rho_2}, where rho_i is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. Probability, Networks and Algorithms (PNA)Software Engineering AbstractWe consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold'. In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold', which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ1, ρ2}, where ρi is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results. IntroductionIn classical queueing networks service stations do not exchange information about their queue lengths. However, in general such communication...

State-dependent importance sampling for a Jackson tandem network

“…Another minor problem is that the function V 2 (x) is not smooth around (α 2 , 0). Specifically for x 2 < δ/γ 2 , the gradient of V 2 (x) is not continuous around the vertical line (x 1 , ·) where the first component satisfies f (x 1 , δ/2γ) = 0 with f (x) as defined in (17). Without going into details, we propose to use any suitable mollification procedure to make V 2 (x) a smooth function, and from now on we will treat V 2 (x) as such.…”

“…[11]; this complication does not play a role when analyzing rare-event probabilities related to the total network population. We expect that the above-mentioned large-deviations heuristic can be rather helpful when analyzing a broad class of networks; see also earlier results in [13] for the model that was introduced in [17], in which the service rate of the first queue depends on the content of the second queue. The proof technique is essentially based on that of Dupuis et al [7], but, as in De Boer and Scheinhardt [4], we have managed to simplify the proofs considerably.…”

Efficient Simulation of a Tandem Queue with Server Slow-down

“…In this paper we study such a model: a tandem queue that consists of two nodes or servers, where, in order to protect the second (downstream) queue from overflow, the first (upstream) server keeps track of queue length at the second server, and lowers its service rate when the second queue is large. In [3] this model was already introduced and the consequences for the first queue were studied, but here our main interest is to determine the probability of overflow in the second queue during a busy cycle. Here we define a busy cycle as the time between two consecutive arrivals to an empty system.…”

State-dependent importance sampling for a slowdown tandem queue

“…Upon combining the fact that W 2 (x) ≥ W 1 (x) − δ for any x ∈D ∪D + (this is shown in the same manner as in Thm. 5.7 of [15]; use (20)), with the monotonicity of γ(x) in both x 1 and x 2 , and using definition (33), it is found that…”

“…For this model only partial results are available, see e.g. [20]. It is noted that the slow-down model is of significant practical interest, as a related mechanism has been proposed e.g.…”