scholarly journals On the Correlation Structure of a Lévy-Driven Queue

2008 ◽  
Vol 45 (4) ◽  
pp. 940-952 ◽  
Author(s):  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Random Variable ◽  
Correlation Structure ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Complementary Distribution ◽  
Completely Monotone ◽  
Server Queue ◽  
Heavy Tailed

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined asr(t) := cov(Q0,Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show thatr(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics ofr(t), for larget, for the cases of light-tailed and heavy-tailed Lévy inputs.

scholarly journals On the Correlation Structure of a Lévy-Driven Queue

2008 ◽  
Vol 45 (04) ◽  
pp. 940-952 ◽  
Author(s):  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Random Variable ◽  
Correlation Structure ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Complementary Distribution ◽  
Completely Monotone ◽  
Server Queue ◽  
Heavy Tailed

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Q t ) t≥0, with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q 0, Q t ) / var(Q 0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0 ∞ r(t)e-ϑt dt. This expression allows us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Lévy inputs.

scholarly journals Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

2011 ◽  
Vol 48 (1) ◽  
pp. 114-130 ◽  
Author(s):  
Peter W. Glynn ◽  
Michel Mandjes
Keyword(s):  
Correlation Function ◽  
Variance Reduction ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Simulation Techniques ◽  
Completely Monotone ◽  
Simulation Based ◽  
Server Queue

In this paper we consider a single-server queue with Lévy input, and, in particular, its workload process (Qt)t≥0, focusing on its correlation structure. With the correlation function defined asr(t):= cov(Q0,Qt) / varQ0(assuming that the workload process is in stationarity at time 0), we first study its transform ∫0∞r(t)e-ϑtdt, both for when the Lévy process has positive jumps and when it has negative jumps. These expressions allow us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior ofr(t) for larget. We then focus on techniques to estimater(t) by simulation. Naive simulation techniques require roughly (r(t))-2runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.

scholarly journals Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

2011 ◽  
Vol 48 (01) ◽  
pp. 114-130 ◽  
Author(s):  
Peter W. Glynn ◽  
Michel Mandjes
Keyword(s):  
Correlation Function ◽  
Variance Reduction ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Simulation Techniques ◽  
Completely Monotone ◽  
Simulation Based ◽  
Server Queue

In this paper we consider a single-server queue with Lévy input, and, in particular, its workload process (Q t ) t≥0, focusing on its correlation structure. With the correlation function defined as r(t):= cov(Q 0, Q t ) / varQ 0 (assuming that the workload process is in stationarity at time 0), we first study its transform ∫0 ∞ r(t)e-ϑt dt, both for when the Lévy process has positive jumps and when it has negative jumps. These expressions allow us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for large t. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly (r(t))-2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.

Finite-pool queueing with heavy-tailed services

2017 ◽  
Vol 54 (3) ◽  
pp. 921-942
Author(s):  
Gianmarco Bet ◽  
Remco van der Hofstad ◽  
Johan S. H. van Leeuwaarden
Keyword(s):  
Head Start ◽  
Stable Process ◽  
Scaling Limit ◽  
Asymptotic Result ◽  
Single Server ◽  
Single Server Queue ◽  
Long Period ◽  
Server Queue ◽  
Service Times ◽  
Heavy Tailed

AbstractWe consider the Δ(i)/G/1 queue, in which a total ofncustomers join a single-server queue for service. Customers join the queue independently after exponential times. We considerheavy-tailedservice-time distributions with tails decaying asx-α, α ∈ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Betet al.(2015)), but then with a Brownian motion instead of an α-stable process.

The total waiting time in a busy period of a stable single-server queue, I.

1969 ◽  
Vol 6 (03) ◽  
pp. 550-564 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Waiting Time ◽  
Queueing Theory ◽  
Busy Period ◽  
Queueing System ◽  
Road Traffic ◽  
Waiting Times ◽  
Random Variable ◽  
Single Server ◽  
Single Server Queue ◽  
Server Queue

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.

The total waiting time in a busy period of a stable single-server queue, I.

1969 ◽  
Vol 6 (3) ◽  
pp. 550-564 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Waiting Time ◽  
Queueing Theory ◽  
Busy Period ◽  
Queueing System ◽  
Road Traffic ◽  
Waiting Times ◽  
Random Variable ◽  
Single Server ◽  
Single Server Queue ◽  
Server Queue

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.

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