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 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 Analysis of Multi-Server Queue with Self-Sustained Servers

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2134
Author(s):  
Alexander Dudin ◽  
Olga Dudina ◽  
Sergei Dudin ◽  
Konstantin Samouylov
Keyword(s):  
Three Dimensional ◽  
Optimal Choice ◽  
Arrival Process ◽  
Queuing Model ◽  
Numerical Result ◽  
Self Sufficiency ◽  
Server Queue ◽  
Markov Arrival ◽  
Markov Arrival Process ◽  
Multi Server

A novel multi-server vacation queuing model is considered. The distinguishing feature of the model, compared to the standard queues, is the self-sufficiency of servers. A server can terminate service and go on vacation independently of the system manager and the overall situation in the system. The system manager can make decisions whether to allow the server to start work after vacation completion and when to try returning some server from a vacation to process customers. The arrival flow is defined by a general batch Markov arrival process. The problem of optimal choice of the total number of servers and the thresholds defining decisions of the manager arises. To solve this problem, the behavior of the system is described by the three-dimensional Markov chain with the special block structure of the generator. Conditions for the ergodicity of this chain are derived, the problem of computation of the steady-state distribution of the chain is discussed. Expressions for the key performance indicators of the system in terms of the distribution of the chain states are derived. An illustrative numerical result is presented.

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).

A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals

1997 ◽  
Vol 34 (3) ◽  
pp. 767-772 ◽  
Author(s):  
John A. Barnes ◽  
Richard Meili
Keyword(s):  
Queueing System ◽  
Mean Shift ◽  
Intensity Function ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Periodic Intensity ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

scholarly journals On optimal stochastic jumps in multi server queue with impatient customers via stochastic control

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ali Delavarkhalafi
Keyword(s):  
Optimal Control ◽  
Complete Information ◽  
Queuing System ◽  
Arrival Process ◽  
Impatient Customers ◽  
Time Intervals ◽  
Poisson Arrival ◽  
Server Queue ◽  
Multi Server ◽  
Request Service

<p style='text-indent:20px;'>In this paper, a queuing system as multi server queue, in which customers have a deadline and they request service from a random number of identical severs, is considered. Indeed there are stochastic jumps, in which the time intervals between successive jumps are independent and exponentially distributed. These jumps will be occurred due to a new arrival or situation change of servers. Therefore the queuing system can be controlled by restricting arrivals as well as rate of service for obtaining optimal stochastic jumps. Our model consists of a single queue with infinity capacity and multi server for a Poisson arrival process. This processes contains deterministic rate <inline-formula><tex-math id="M1">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and exponential service processes with <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> rate. In this case relevant customers have exponential deadlines until beginning of their service. Our contribution is to extend the Ittimakin and Kao's results to queueing system with impatient customers. We also formulate the aforementioned problem with complete information as a stochastic optimal control. This optimal control law is found through dynamic programming.</p>

scholarly journals Exact simulation of the extrema of stable processes

2019 ◽  
Vol 51 (4) ◽  
pp. 967-993
Author(s):  
Jorge I. González Cázares ◽  
Aleksandar Mijatović ◽  
Gerónimo Uribe Bravo
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Stable Process ◽  
Stable Processes ◽  
Time Interval ◽  
Simulation Algorithm ◽  
Finite Time Interval ◽  
Exact Simulation ◽  
The Past ◽  
Coupling From The Past

AbstractWe exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes [27]) and use it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse the performance of the algorithm numerically (the code, written in Julia 1.0, is available on GitHub).

scholarly journals Queues, stores, and tableaux

2005 ◽  
Vol 42 (04) ◽  
pp. 1145-1167 ◽  
Author(s):  
Moez Draief ◽  
Jean Mairesse ◽  
Neil O'Connell
Keyword(s):  
Stability Condition ◽  
Young Tableau ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Departure Process ◽  
Transient Behaviour ◽  
Server Queue ◽  
The Stability ◽  
First In First Out

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (1) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

On ross's conjectures about queues with non-stationary poisson arrivals

1982 ◽  
Vol 19 (01) ◽  
pp. 245-249 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Poisson Process ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Average Delay ◽  
Server Queue ◽  
Poisson Arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

Sign in / Sign up

Export Citation Format

Share Document