Stationary analysis of a single queue with remaining service time-dependent arrivals

Download

index.pdf

Date

2018-02-01

Author

Legros, Benjamin
Sezer, Ali Devin

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

154
views

126
downloads

We study a generalization of the M / G / 1 system (denoted by rM / G / 1) with independent and identically distributed service times and with an arrival process whose arrival rate depends on the remaining service time r of the current customer being served. We derive a natural stability condition and provide a stationary analysis under it both at service completion times (of the queue length process) and in continuous time (of the queue length and the residual service time). In particular, we show that the stationary measure of queue length at service completion times is equal to that of a corresponding M / G / 1 system. For , we show that the continuous time stationary measure of the rM / G / 1 system is linked to the M / G / 1 system via a time change. As opposed to the M / G / 1 queue, the stationary measure of queue length of the rM / G / 1 system at service completions differs from its marginal distribution under the continuous time stationary measure. Thus, in general, arrivals of the rM / G / 1 system do not see time averages. We derive formulas for the average queue length, probability of an empty system and average waiting time under the continuous time stationary measure. We provide examples showing the effect of changing the reshaping function on the average waiting time.

Subject Keywords

Residual service time dependent arrivals, Reshaping function, Queueing systems, Performance evaluation, Piecewise deterministic processes

URI

https://hdl.handle.net/11511/30928

Journal

QUEUEING SYSTEMS

DOI

https://doi.org/10.1007/s11134-017-9552-z

Collections

Graduate School of Applied Mathematics, Article

Suggestions

OpenMETU
Core

Exit probabilities of markov modulated constrained random walks Başoğlu Kabran, Fatma; Sezer, Ali Devin; Department of Financial Mathematics (2018) Let X be the constrained random walk on Z2+ with increments (0, 0), (1, 0), (−1, 1), (0, −1) whose jump probabilities are determined by the state of a finite state Markov chain M. X represents the lengths of two queues of customers (or packets, tasks, etc.) waiting for service from two servers working in tandem; the arrival of customers occur with rate λ(Mk), service takes place at rates μ1(Mk), and μ2(Mk) where Mk denotes the current state of the Markov chain M. We assume that the average arrival rate is l...
Repetition support and mining cyclic patterns Toroslu, İsmail Hakkı (Elsevier BV, 2003-10-01) For customer transaction database, the support (customer support) of the sequential pattern is defined as the fraction of the customers supporting the sequence. We define new forms of the mining patterns, called cyclic patterns, as an extension to the sequential pattern mining by introducing a new parameter, the repetition support. For customer transaction database, the repetition support specifies the minimum number of repetitions of the patterns in each customer transaction sequence. Repeated patterns can...
Approximation of excessive backlog probabilities of two parallel queues Ünlü, Kamil Demirberk; Sezer, Ali Devin (2018-07-25) Let X be the constrained random walk on Z 2 + with increments (1, 0), (−1, 0), (0, 1) and (0, −1) representing the lengths at service completion times of two queues with exponentially distributed interarrival and service times running in parallel. Denote the arrival and service rates by λ i , µ i , i = 1, 2; we assume λ i < µ i , i = 1, 2, i.e., X is assumed stable. Without loss of generality we assume ρ 1 ≥ ρ 2. Let τ n be the first time X hits the line ∂A n = {x ∈ Z 2 : x(1)+x(2) = n}, i.e., when the sum ...
Dependability design for distributed real-time systems with broadcast communication / Kartal, Yusuf Bora; Schmidt, Şenan Ece; Department of Electrical and Electronics Engineering (2014) The operation of distributed systems relies on the timely exchange of message data via dependable communication networks. Previous works suggest hardware redundancy for potential faults in the underlying network infrastructure to achieve dependability. However, software faults and faults that cannot be resolved on the hardware level are not considered in the existing literature. This work proposes a new method for software fault-tolerant communication in distributed real-time systems with communication netw...
Cost aware TCP scheduler for bandwidth aggregation Ağırbaş, Serhat; Koçyiğit, Altan; Sevgi, Cüneyt; Department of Information Systems (2018) Constant bit rate, time sensitive data delivery is needed for many network applications and these applications usually require high throughput and less variable delay. When such applications run on mobile devices, the bandwidth available and the other characteristics of the primary network connection may not be sufficient to provide necessary quality of service. On the other hand, most of the mobile devices are multihomed that is they are equipped with more than one network interface, hence they can be conn...

Citation Formats

B. Legros and A. D. Sezer, “Stationary analysis of a single queue with remaining service time-dependent arrivals,” QUEUEING SYSTEMS, pp. 139–165, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30928.