scholarly journals Importance sampling for Markovian tandem queues using subsolutions: exploring the possibilities

SIMULATION ◽  
2021 ◽  
Vol 97 (12) ◽  
pp. 849-866
Author(s):  
Anne Buijsrogge ◽  
Pieter-Tjerk de Boer ◽  
Werner R W Scheinhardt
Keyword(s):  
Importance Sampling ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Asymptotically Efficient ◽  
Number Of Customers ◽  
Busy Cycle

We consider importance sampling simulation for estimating the probability of reaching large total number of customers in an [Formula: see text] tandem queue, during a busy cycle of the system. Our main result is a procedure for obtaining a family of asymptotically efficient changes of measure based on subsolutions. We explicitly show these families for two-node tandem queues and we find that there exist more asymptotically efficient changes of measure based on subsolutions than currently available in literature.

scholarly journals ON STATE-INDEPENDENT IMPORTANCE SAMPLING FOR THE GI|GI|1 TANDEM QUEUE1

2018 ◽  
Vol 34 (1) ◽  
pp. 131-156 ◽  
Author(s):  
Anne Buijsrogge ◽  
Pieter-Tjerk de Boer ◽  
Werner R.W. Scheinhardt
Keyword(s):  
Importance Sampling ◽  
Arrival Process ◽  
Change Of Measure ◽  
Tandem Queue ◽  
Efficient Estimator ◽  
Asymptotically Efficient Estimator ◽  
High Level ◽  
Asymptotically Efficient ◽  
Sufficient Precision ◽  
Busy Cycle

In this paper, we consider a d-node GI|GI|1 tandem queue with i.i.d. inter-arrival process and service processes that are independent of each other. Our main interest is to estimate the probability to reach a high level N in a busy cycle of the system using simulation. As crude simulation does not give a sufficient precision in reasonable time, we use importance sampling. We introduce a method to find a state-independent change of measure and we show that this is equivalent to a change of measure that was earlier, but implicitly, described by Parekh and Walrand [8]. We also show that this change of measure is the only exponential state-independent change of measure that may result in an asymptotically efficient estimator. Lastly, we provide necessary conditions for this state-independent change of measure to give an asymptotically efficient estimator.

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.

Regeneration in tandem queues with multiserver stations

1988 ◽  
Vol 25 (02) ◽  
pp. 391-403 ◽  
Author(s):  
Karl Sigman
Keyword(s):  
Markov Chain ◽  
Renewal Process ◽  
Single Server ◽  
Tandem Queues ◽  
Random Assignment ◽  
Tandem Queue ◽  
Ergodic Markov Chain ◽  
Fifo Queue ◽  
Harris Chain ◽  
Server System

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (03) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a generalGI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

scholarly journals Importance sampling for non-Markovian tandem queues using subsolutions

2019 ◽  
Vol 93 (1-2) ◽  
pp. 31-65
Author(s):  
Anne Buijsrogge ◽  
Pieter-Tjerk de Boer ◽  
Werner R. W. Scheinhardt
Keyword(s):  
Importance Sampling ◽  
Tandem Queues

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (3) ◽  
pp. 815-829 ◽  
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

scholarly journals Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012 ◽  
Vol 44 (04) ◽  
pp. 1173-1196
Author(s):  
Hock Peng Chan ◽  
Shaojie Deng ◽  
Tze-Leung Lai
Keyword(s):  
Random Walks ◽  
Importance Sampling ◽  
Rare Event ◽  
Sequential Importance Sampling ◽  
New Approach ◽  
Martingale Representation ◽  
Event Simulation ◽  
Markov Random ◽  
Heavy Tailed ◽  
Asymptotically Efficient

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

scholarly journals On asymptotically efficient simulation of large deviation probabilities

2005 ◽  
Vol 37 (2) ◽  
pp. 539-552 ◽  
Author(s):  
A. B. Dieker ◽  
M. Mandjes
Keyword(s):  
Importance Sampling ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Sampling Distribution ◽  
Efficient Simulation ◽  
Deviation Principle ◽  
Asymptotically Efficient Estimator ◽  
Asymptotically Efficient ◽  
Necessary And Sufficient

Let {νε, ε>0} be a family of probabilities for which the decay is governed by a large deviation principle, and consider the simulation of νε0(A) for some fixed measurable set A and some ε0>0. We investigate the circumstances under which an exponentially twisted importance sampling distribution yields an asymptotically efficient estimator. Varadhan's lemma yields necessary and sufficient conditions, and these are shown to improve on certain conditions of Sadowsky. This is illustrated by an example to which Sadowsky's conditions do not apply, yet for which an efficient twist exists.

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