scholarly journals Exact simulation of the stationary distribution of the FIFO M/G/c queue

2011 ◽  
Vol 48 (A) ◽  
pp. 209-213 ◽  
Author(s):  
Karl Sigman
Keyword(s):  
Stationary Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Processor Sharing ◽  
Exact Simulation ◽  
Jackson Networks ◽  
The Past ◽  
Customer Delay ◽  
First In First Out ◽  
General Method

We present an exact simulation algorithm for the stationary distribution of the customer delayDfor first-in–first-out (FIFO) M/G/cqueues in which ρ=λ/μ<1. We assume that the service time distributionG(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distributionGe(x)=μ∫0xP(S>y)dyare such that samples of them can be simulated. We further assume thatGhas a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which isD. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

scholarly journals Exact simulation of the stationary distribution of the FIFO M/G/c queue

2011 ◽  
Vol 48 (A) ◽  
pp. 209-213 ◽  
Author(s):  
Karl Sigman
Keyword(s):  
Stationary Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Processor Sharing ◽  
Exact Simulation ◽  
Jackson Networks ◽  
The Past ◽  
Customer Delay ◽  
First In First Out ◽  
General Method

We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

scholarly journals Exact sampling for some multi-dimensional queueing models with renewal input

2019 ◽  
Vol 51 (4) ◽  
pp. 1179-1208 ◽  
Author(s):  
Jose Blanchet ◽  
Yanan Pei ◽  
Karl Sigman
Keyword(s):  
Stationary Distribution ◽  
Arrival Process ◽  
Exact Simulation ◽  
The Past ◽  
Coupling From The Past ◽  
Server Queue ◽  
Negative Drift ◽  
Poisson Arrivals ◽  
First In First Out ◽  
Multi Server

AbstractUsing a result of Blanchet and Wallwater (2015) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (i.i.d.) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in–first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are i.i.d.: the FIFO GI/GI/c queue with $ 2 \leq c \lt \infty$ . Our method utilizes dominated coupling from the past (DCFP) as well as the random assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015). We also consider the models in continuous time, and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/ $\infty$ model. Finally, we even show how to handle fork–join queues, in which each arriving customer brings c jobs, one for each server.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (3) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

On queues with periodic Poisson input

1981 ◽  
Vol 18 (04) ◽  
pp. 889-900 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Direct Proof ◽  
Poisson Model ◽  
Fixed Time ◽  
Time Of Day ◽  
Asymptotic Distributions ◽  
Single Server ◽  
Service Time Distribution ◽  
Asymptotic Results ◽  
Poisson Input

This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takács. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed ‘time of day' coincides with the distribution for the supremum, over the time horizon [0,∞), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.

scholarly journals Bio-Based Polymer Electrolytes for Electrochemical Devices: Insight into the Ionic Conductivity Performance

Materials ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 838 ◽  
Author(s):  
Marwah Rayung ◽  
Min Min Aung ◽  
Shah Christirani Azhar ◽  
Luqman Chuah Abdullah ◽  
Mohd Sukor Su’ait ◽  
...  
Keyword(s):  
Ionic Conductivity ◽  
Polymer Electrolyte ◽  
Polymer Electrolytes ◽  
Future Prospects ◽  
Electrolyte System ◽  
The Past ◽  
Device Applications ◽  
Electrochemical Devices ◽  
Insight Into ◽  
General Method

With the continuing efforts to explore alternatives to petrochemical-based polymers and the escalating demand to minimize environmental impact, bio-based polymers have gained a massive amount of attention over the last few decades. The potential uses of these bio-based polymers are varied, from household goods to high end and advanced applications. To some extent, they can solve the depletion and sustainability issues of conventional polymers. As such, this article reviews the trends and developments of bio-based polymers for the preparation of polymer electrolytes that are intended for use in electrochemical device applications. A range of bio-based polymers are presented by focusing on the source, the general method of preparation, and the properties of the polymer electrolyte system, specifically with reference to the ionic conductivity. Some major applications of bio-based polymer electrolytes are discussed. This review examines the past studies and future prospects of these materials in the polymer electrolyte field.

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