Some Inequalities for the Queue GI/G/1

Kingmán, J. F. C.

doi:10.2307/2333966

Cited by 73 publications

(58 citation statements)

References 4 publications

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“…We now wish to apply our results to estimating when fx < 0 and Ee~$ w = d(s) exists for at least -c x < s ^ 0, q > 0. Our estimate extends that obtained in § II.5 of [4] for the difference EM oo -EM n , and also the work of Kingman [6].…”

Section: F^n -E[a(yovn)e»rw*] = F A(x)e»*dp(z N (S 0 ) 5= X)supporting

confidence: 91%

An expansion related to the central limit theorem

Heathcote

1969

J. Aust. Math. Soc.

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Section: F^n -E[a(yovn)e»rw*] = F A(x)e»*dp(z N (S 0 ) 5= X)supporting

confidence: 91%

An expansion related to the central limit theorem

Heathcote

1969

J. Aust. Math. Soc.

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“…It is easy to see that the product ofT i and j∈Ni πj λj 1+λj stays the same if the relative values of {π j : j ∈ N i } are kept fixed. Since the variance of the time link i gets the medium or has an outage is finite, it is easy to argue using Kingman's bounds [8], [9], [10] that the mean delay will be finite for all average arrival rates strictly less thans i , if the variance of the arrival process is also finite. From Theorem 2 this is an upper-bound on the mean delay for the real system.…”

Section: T) Is Given By τ (I − T)mentioning

confidence: 99%

Delay performance of CSMA in networks with bounded degree conflict graphs

Subramanian

Alanyali

2011

2011 IEEE International Symposium on Information Theory Proceedings

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Abstract-We analyze packet delay in CSMA-based random access schemes in networks under the protocol interference model. Using a stochastic coupling argument we identify a subset of the throughput-region where queue lengths can be bounded uniformly for all network sizes. This conclusion provides a throughput-region of interest for delay sensitive applications and suggests that delay bounds based on mixing time analyses may be loose.

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“…and Gamma distributed. Given that all queues become independent Γ/GI/1 queues, classic heavy-tra c analysis immediately applies to derive useful approximation and insights; see, e.g., [23,16].…”

Section: Heterogeneous Queuesmentioning

confidence: 99%

The price of forgetting in parallel and non-observable queues

Anselmi¹,

Gaujal²

2011

Performance Evaluation

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We consider a broker-based network of non-observable parallel queues and analyze the minimum expected response time and the optimal routing policy when the broker has the memory of its previous routing decisions. We provide lower bounds on the minimum response time by means of convex programming that are tight, as follows by a numerical comparison with a proposed routing scheme. The Price of Forgetting (PoF), the ratio between the minimum response times achieved by a probabilistic broker and a broker with memory, is shown to be unbounded or arbitrarily close to one depending on the coe cient of variation of the service time distributions. In the case of exponential service times, the PoF is bounded from above by two, which is tight in heavy-tra c, and independent of the network size and heterogeneity. These properties yield a simple engineering product-form approximating tightly the minimum response time. Finally, we put our results in the context of game theory revisiting the Price of Anarchy (PoA) of parallel queues: It can be decomposed into the product of the PoA achieved by a probabilistic broker (already well understood) and the PoF. IntroductionIn the context of *-computing and manufacturing systems, users submit jobs without knowing which machine will handle their execution. A central broker is usually in charge of distributing incoming jobs to a set of resources to optimize some utility or cost function. The mean response time (simply response time in the following), the expected time it takes for a job to join the system and return to its issuer, is an important metric that is usually taken into account to achieve a better exploitation of resources and improve the average quality of service.The problem of nding the routing strategy (or policy) that minimizes response time and the resulting response time itself are well-known problem in queueing theory and the literature is overwhelming. There are two main classes of routing strategies: Closed-loop policies, where the broker knows the state of each resource (number of jobs), and open-loop policies, where the broker does not know their state. Computing the best closed-loop policy is known to be a hard problem and a lot of work has been devoted to the computation of good approximations; see, e.g., [9,27, 33] and the references therein.The focus of this paper is on the open-loop case, and more precisely on o -line policies. Indeed, the state of the queues may not to be known on-line to the broker because of a number of reasons whose e ects become worse and worse as the network size increases: i) Communicating the system state to the broker increases the network load, ii) the information received by the broker can be out of date, and iii) the synchronization that results from knowing the system state can degrade the performance [26]. A celebrated case is when the broker only knows the service time distributions of the network resources (or queues in the following) and dispatches jobs to queues according to an i.i.d. probabilistic law. The probabili...

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Some Inequalities for the Queue GI/G/1

Cited by 73 publications

References 4 publications

An expansion related to the central limit theorem

An expansion related to the central limit theorem

Delay performance of CSMA in networks with bounded degree conflict graphs

The price of forgetting in parallel and non-observable queues

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