Mean queue size in a queue with discrete autoregressive arrivals of order p

Kim, Jeongsim; Kim, Bara; Sohraby, K.

doi:10.1007/s10479-008-0318-1

Cited by 7 publications

(14 citation statements)

References 19 publications

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“…The description of the general DAR( n ) sources can be found in . The DAR(2) source, denoted { A t , t = 0,1, … }, is a special case and can be defined by

A_{t} MathClass-rel= (1 MathClass-bin- ν_{t}) B_{t} MathClass-bin+ ν_{t} A_{t MathClass-bin- Φ_{t}} MathClass-punc,

where { ν t }is a sequence of independent Bernoulli random variables with

P (ν_{t} MathClass-rel= 1) MathClass-rel= p

, { B t }is another sequence of nonnegative independent and identically distributed random variables.…”

Section: Slower Decay Rate In the Autocorrelation Function Of Dar(2)mentioning

confidence: 61%

“…The analysis in gives a formula for the mean queue size of the aforementioned system. Because the performance measures being considered are both the first‐ and second‐order statistics of the queue size X t , we take an alternative approach and follow to derive the closed‐form results for the first two moments:

L MathClass-rel= E [X_{t}] and L^{(2)} MathClass-rel= E [X_{t}^{2}]

.…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…This simplifies the analysis and leads to the following lemma, which we will need in the subsequent derivations. A similar point was made in . Lemma Under the condition , we have

E ([], A_{t MathClass-bin- 1}^{k} U_{t}) MathClass-rel= E ([], A_{t MathClass-bin- 1}^{k}) MathClass-punc,

where

A_{t MathClass-bin- 1}^{k}

denotes the k th power of A t −1 . Proof According to and , if A t −1 ≠ 0, then the batch size is at least 2 implying X t −1 , X t > 0 and U t −1 = U t = 1.…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…This enables us to express α 1 as

α_{1} MathClass-rel= λ (L MathClass-bin- λ) MathClass-bin+ \frac{p ((), 1 MathClass-bin- p ϕ_{2}^{2}) E [A^{2}]}{(1 MathClass-bin- p) (1 MathClass-bin- p ϕ_{2})} MathClass-bin+ \frac{p ϕ_{2} λ^{2}}{1 MathClass-bin- p ϕ_{2}} MathClass-bin- \frac{p (1 MathClass-bin+ ϕ_{2}) λ}{1 MathClass-bin- p} MathClass-punc.

Inserting α 1 back to , we obtain the formula for the mean queue size in the following. This agrees with the existing result in , but our derivation here is much simpler and the result given below is neater. Proposition For the DAR(2)/D/1, the mean queue size is given by

L MathClass-rel= \frac{1}{1 MathClass-bin- λ} ([], \frac{E [A^{2}] MathClass-bin- 2 λ^{2} MathClass-bin+ λ}{2} MathClass-bin+ \frac{p ((), 1 MathClass-bin- p ϕ_{2}^{2}) E [A^{2}]}{(1 MathClass-bin- p) (1 MathClass-bin- p ϕ_{2})} MathClass-bin+ \frac{p ϕ_{2} λ^{2}}{1 MathClass-bin- p ϕ_{2}} MathClass-bin- p (1 MathClass-bin+ ϕ_{2})

…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…The description of the general DAR(n) sources can be found in [8,9]. The DAR(2) source, denoted fA t ; t D 0; 1; : : :g, is a special case and can be defined by…”

Section: Slower Decay Rate In the Autocorrelation Function Of Dar(2)mentioning

confidence: 99%

See 4 more Smart Citations

Second‐order performance analysis of discrete‐time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter

Miao

Lee

2012

Appl Stoch Models Bus & Ind

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Discrete autoregressive process of order 1 (DAR(1)) has been used as a popular stochastic model for correlated traffic sources because it parsimoniously uses a single parameter to capture the desirable correlation structure. In contrast with DAR(1), discrete autoregressive process of order 2 (DAR(2)) uses one more parameter to provide a much richer pattern in the autocorrelation function and is able to capture slower decay rate and longer memory. To investigate how the additional traffic parameter in DAR(2) influences the queueing performance, this paper provides an analysis of the discrete‐time DAR(2)/D/1 queue. The performance measures concerned are the mean and second‐order statistics of queue size, which are both important in the queueing systems seen in telecommunication networks. Under a mild condition, these performance indices are derived in closed form that allows for efficient computing. An approximate version of these results is also developed to relax the condition and cover more general sources, and both versions serve as a simple tool set for performance evaluation. The numerical examples use this tool to demonstrate that the DAR(2) source may cause up to 30% poorer performance than DAR(1) when the traffic is heavy, bursty, and highly correlated. This indicates that the effect from slower decay rate in autocorrelation is not negligible and using the extra parameter is necessary particularly when the queue is heavily loaded with correlated traffic. Copyright © 2012 John Wiley & Sons, Ltd.

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“…The description of the general DAR( n ) sources can be found in . The DAR(2) source, denoted { A t , t = 0,1, … }, is a special case and can be defined by

A_{t} MathClass-rel= (1 MathClass-bin- ν_{t}) B_{t} MathClass-bin+ ν_{t} A_{t MathClass-bin- Φ_{t}} MathClass-punc,

where { ν t }is a sequence of independent Bernoulli random variables with

P (ν_{t} MathClass-rel= 1) MathClass-rel= p

, { B t }is another sequence of nonnegative independent and identically distributed random variables.…”

Section: Slower Decay Rate In the Autocorrelation Function Of Dar(2)mentioning

confidence: 61%

L MathClass-rel= E [X_{t}] and L^{(2)} MathClass-rel= E [X_{t}^{2}]

.…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…This simplifies the analysis and leads to the following lemma, which we will need in the subsequent derivations. A similar point was made in . Lemma Under the condition , we have

E ([], A_{t MathClass-bin- 1}^{k} U_{t}) MathClass-rel= E ([], A_{t MathClass-bin- 1}^{k}) MathClass-punc,

where

A_{t MathClass-bin- 1}^{k}

denotes the k th power of A t −1 . Proof According to and , if A t −1 ≠ 0, then the batch size is at least 2 implying X t −1 , X t > 0 and U t −1 = U t = 1.…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…This enables us to express α 1 as

α_{1} MathClass-rel= λ (L MathClass-bin- λ) MathClass-bin+ \frac{p ((), 1 MathClass-bin- p ϕ_{2}^{2}) E [A^{2}]}{(1 MathClass-bin- p) (1 MathClass-bin- p ϕ_{2})} MathClass-bin+ \frac{p ϕ_{2} λ^{2}}{1 MathClass-bin- p ϕ_{2}} MathClass-bin- \frac{p (1 MathClass-bin+ ϕ_{2}) λ}{1 MathClass-bin- p} MathClass-punc.

L MathClass-rel= \frac{1}{1 MathClass-bin- λ} ([], \frac{E [A^{2}] MathClass-bin- 2 λ^{2} MathClass-bin+ λ}{2} MathClass-bin+ \frac{p ((), 1 MathClass-bin- p ϕ_{2}^{2}) E [A^{2}]}{(1 MathClass-bin- p) (1 MathClass-bin- p ϕ_{2})} MathClass-bin+ \frac{p ϕ_{2} λ^{2}}{1 MathClass-bin- p ϕ_{2}} MathClass-bin- p (1 MathClass-bin+ ϕ_{2})

…”

Section: Performance Analysis Of the Dar(2)/d/1 Queuementioning

confidence: 99%

“…The description of the general DAR(n) sources can be found in [8,9]. The DAR(2) source, denoted fA t ; t D 0; 1; : : :g, is a special case and can be defined by…”

Section: Slower Decay Rate In the Autocorrelation Function Of Dar(2)mentioning

confidence: 99%

See 3 more Smart Citations

Second‐order performance analysis of discrete‐time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter

Miao

Lee

2012

Appl Stoch Models Bus & Ind

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Queueing models for the analysis of communication systems

Bruneel

Fiems

Walraevens

et al. 2014

TOP

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Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays

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On the Variances of System Size and Sojourn Time in a Discrete-Time Dar(1)/D/1 Queue

Miao¹,

Chen²

2011

Prob. Eng. Inf. Sci.

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We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

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Mean queue size in a queue with discrete autoregressive arrivals of order p

Cited by 7 publications

References 19 publications

Second‐order performance analysis of discrete‐time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter

Second‐order performance analysis of discrete‐time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter

Queueing models for the analysis of communication systems

On the Variances of System Size and Sojourn Time in a Discrete-Time Dar(1)/D/1 Queue

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