scholarly journals Large Fork-Join Queues with Nearly Deterministic Arrival and Service Times

2021 ◽  
Author(s):  
Dennis Schol ◽  
Maria Vlasiou ◽  
Bert Zwart
Keyword(s):  
Extreme Value Theory ◽  
Queue Length ◽  
Value Theory ◽  
Initial Number ◽  
Diffusion Approximations ◽  
Fluid Limit ◽  
Maximum Queue Length ◽  
Queue Lengths ◽  
Service Times ◽  
And Diffusion

In this paper, we study an N server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as [Formula: see text]. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (03) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

scholarly journals Queue length asymptotics for the multiple-server queue with heavy-tailed Weibull service times

2019 ◽  
Vol 93 (3-4) ◽  
pp. 195-226
Author(s):  
Mihail Bazhba ◽  
Jose Blanchet ◽  
Chang-Han Rhee ◽  
Bert Zwart
Keyword(s):  
Queue Length ◽  
Sample Path ◽  
Continuous Mapping ◽  
Asymmetric Structure ◽  
Large Deviations Principle ◽  
Physical Insight ◽  
Queue Lengths ◽  
Service Times ◽  
Heavy Tailed ◽  
Sample Path Large Deviations

Abstract We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a recently developed sample path large-deviations principle for Lévy processes and random walks, following a continuous mapping approach. Also, we identify and solve a key variational problem which provides physical insight into the way a large queue length occurs. In contrast to the regularly varying case, we observe several subtle features such as a non-trivial trade-off between the number of big jobs and their sizes and a surprising asymmetric structure in asymptotic job sizes leading to congestion.

scholarly journals Rare events of transitory queues

2017 ◽  
Vol 54 (3) ◽  
pp. 943-962
Author(s):  
Harsha Honnappa
Keyword(s):  
Rare Events ◽  
Rare Event ◽  
Random Variables ◽  
General Distribution ◽  
Diffusion Approximations ◽  
Large Deviations Principle ◽  
Exponential Random Variables ◽  
Service Times ◽  
Ordered Statistics ◽  
And Diffusion

AbstractWe study the rare-event behavior of the workload process in a transitory queue, where the arrival epochs (or 'points') of a finite number of jobs are assumed to be the ordered statistics of independent and identically distributed (i.i.d.) random variables. The service times (or 'marks') of the jobs are assumed to be i.i.d. random variables with a general distribution, that are jointly independent of the arrival epochs. Under the assumption that the service times are strictly positive, we derive the large deviations principle (LDP) satisfied by the workload process. The analysis leverages the connection between ordered statistics and self-normalized sums of exponential random variables to establish the LDP. In this paper we present the first analysis of rare events in transitory queueing models, supplementing prior work that has focused on fluid and diffusion approximations.

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