Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

1989 ◽  
Vol 4 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Kim L. Ong ◽  
Michael R. Taaffe
Keyword(s):  
Nonstationary Queues ◽  
Poisson Arrivals

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

scholarly journals Performance Evaluation of Cloud Data Centers with Batch Task Arrivals

2013 ◽  
pp. 199-223 ◽  
Author(s):  
Hamzeh Khazaei ◽  
Jelena Mišić ◽  
Vojislav B. Mišić
Keyword(s):  
Performance Evaluation ◽  
Queue Length ◽  
Blocking Probability ◽  
General Distribution ◽  
Cloud Data ◽  
Wide Range ◽  
Accurate Performance ◽  
Cloud Data Centers ◽  
Poisson Arrivals

Accurate performance evaluation of cloud computing resources is a necessary prerequisite for ensuring that Quality of Service (QoS) parameters remain within agreed limits. In this chapter, the authors consider cloud centers with Poisson arrivals of batch task requests under total rejection policy; task service times are assumed to follow a general distribution. They describe a new approximate analytical model for performance evaluation of such systems and show that important performance indicators such as mean request response time, waiting time in the queue, queue length, blocking probability, probability of immediate service, and probability distribution of the number of tasks in the system can be obtained in a wide range of input parameters.

A martingale approach to the PASTA property

1993 ◽  
Vol 30 (01) ◽  
pp. 252-257
Author(s):  
Michael Scheutzow
Keyword(s):  
Poisson Process ◽  
Queueing Theory ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Boundedness Condition ◽  
Martingale Approach ◽  
Large Numbers ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Image Position

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L 2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

scholarly journals Relating Time and Customer Averages for Queues Using ‘forward’ Coupling from the Past

2008 ◽  
Vol 45 (02) ◽  
pp. 568-574
Author(s):  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Queueing System ◽  
Steady State Distribution ◽  
State Distribution ◽  
The Past ◽  
Coupling From The Past ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Time Average ◽  
New Better Than Used ◽  
Better Than

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

Sign in / Sign up

Export Citation Format

Share Document