Fluid and diffusion limits for transient sojourn times of processor sharing queues with time varying rates

Hampshire, Robert C.; Massey, William A.

doi:10.1007/s11134-006-7584-x

Cited by 23 publications

(17 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The PS interpretation of uniform acceleration is discussed in Hampshire et al [17]. Through the theory of strong approximations (see Kolmós et al [29], Ethier and Kurtz [8]), we can also apply these same uniform acceleration asymptotics directly to the random sample path behavior of this queueing process and obtain…”

Section: Service Delay Modelsmentioning

confidence: 90%

Dynamic Optimization with Applications to Dynamic Rate Queues

Hampshire

Heinz

Massey

2010

Risk and Optimization in an Uncertain World

Self Cite

View full text Add to dashboard Cite

This tutorial presents recent developments in the management of communications services and applies broadly to services involving the leasing of shared resources. These problems are more realistically modeled by queues with time-varying rates or more simply, dynamic rate queues. We first provide a review and summary of relevant results for various fundamental dynamic rate queues. The focus here is on approximations of these queueing models by low-dimensional dynamical systems. The dynamic optimization of constrained dynamical systems is based on the calculus of variations and its various incarnations over the past three centuries. We discuss these methods in the context of Lagrangians, Hamiltonians, and Bellman value functions. Finally, we provide examples where we apply these optimization techniques to dynamic rate queues motivated by communications decision problems.Keywords Bellman value function; calculus of variations; conservation principles; differential equations; dynamical systems; Hamiltonian; Lagrangian; Lagrange multipliers; Legendre transforms; opportunity costs; optimal control; Poisson brackets The Operations of Communication Systems and Services Four Canonical IssuesWe begin by discussing the operational issues motivated by the communications industry. This is the business sector that created the telephone network in the beginning of the 20th century and gave birth to the mathematical field of queueing theory. Around the middle of the 20th century, or the 1960s, queueing theory was found to have applications to the communication of computer systems. The mathematics of queueing has taken on a life of its own since then and has applications to all services involving the leasing of shared resources. A manager of a communications business has a continual interest in offering various services as efficiently as possible in terms of the company resources. We can express these concerns in terms of four canonical issues: performance, pricing, provisioning, and prioritization. The first one is that of service performance. These are metrics that predict the availability of resources needed for service to the typical, newly arriving customer. This is expressed in terms of the past demand and usage of these resources by previous customers.The second issue is service pricing. If the price is too high, then no one may use the service. If the price is too low, then the cost of the resources used by the business providing the service may exceed the revenue obtained. A decision must be made as to what is the most strategic price in terms of optimizing revenue without compromising some target level of service performance. If the general strategy for a decision can be expressed in terms of a finite set of simple steps, then we call this a policy. Since the larger goal of operations 208

show abstract

Section: Service Delay Modelsmentioning

confidence: 90%

Dynamic Optimization with Applications to Dynamic Rate Queues

Hampshire

Heinz

Massey

2010

Risk and Optimization in an Uncertain World

Self Cite

View full text Add to dashboard Cite

show abstract

“…This note, based on work in Hampshire [4], is an extension of [5]. Here we chronicle a phenomenon that can only be observed in a processor sharing system with time varying rates.…”

Section: Introductionmentioning

confidence: 87%

“…The work of Hampshire, Harchol-Balter and Massey [5] derives fluid and diffusion limit theorems for the sojourn time of a virtual customer in an M t /M/1 processor sharing queue with time varying or non-homogeneous, Poisson arrival rates. This note, based on work in Hampshire [4], is an extension of [5].…”

Section: Introductionmentioning

confidence: 99%

“…This property was discovered while correcting a proof in Hampshire, Harchol-Balter and Massey [5]. If the arrival rate for processor sharing queue has unbounded growth over time, then it is possible for the number of customers in a processor sharing queue to grow so quickly that a newly entering job never finishes.…”

mentioning

confidence: 98%

“…We discuss the use of such a concept and develop some of its properties. This short article serves both as errata for [5] and as documentation of a characteristic feature for some processor sharing queues with time varying rates. …”

mentioning

confidence: 99%

See 2 more Smart Citations

A note on the event horizon for a processor sharing queue

Hampshire

Massey

2008

Queueing Syst

Self Cite

View full text Add to dashboard Cite

In this note we identify a phenomenon for processor sharing queues that is unique to ones with time-varying rates. This property was discovered while correcting a proof in Hampshire, Harchol-Balter and Massey [5]. If the arrival rate for processor sharing queue has unbounded growth over time, then it is possible for the number of customers in a processor sharing queue to grow so quickly that a newly entering job never finishes. We define the minimum size for such a job to be the event horizon for a processor sharing queue. We discuss the use of such a concept and develop some of its properties. This short article serves both as errata for [5] and as documentation of a characteristic feature for some processor sharing queues with time varying rates.

show abstract

Additive functionals with application to sojourn times in infinite-server and processor sharing systems

Brandt

2012

Queueing Syst

View full text Add to dashboard Cite

Fluid and diffusion limits for transient sojourn times of processor sharing queues with time varying rates

Cited by 23 publications

References 25 publications

Dynamic Optimization with Applications to Dynamic Rate Queues

Dynamic Optimization with Applications to Dynamic Rate Queues

A note on the event horizon for a processor sharing queue

Additive functionals with application to sojourn times in infinite-server and processor sharing systems

Contact Info

Product

Resources

About