Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control

Lee, Duan-Shin; Li, San-Qi

doi:10.1016/0166-5316(92)90067-q

Cited by 13 publications

(11 citation statements)

References 29 publications

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“…Some examples are (in chronological order) [12,13,15,14,16]. Here, [12] studies the transient behavior of a Markov-modulated Poisson arrival queue with multiple exponential servers under overload control. In [13], a M AP/M/2 system with two classes of customers is considered, where one type of customers requires only one server and the other type needs both servers.…”

Section: Continuous-time Modelsmentioning

confidence: 99%

Analysis of a discrete-time single-server queue with an occasional extra server

Bruneel

Wittevrongel

2017

Performance Evaluation

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We consider a discrete-time queueing system having two distinct servers: one server, the "regular" server, is permanently available, while the second server, referred to as the "extra" server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random "up-periods", whereas it is unavailable during random "down-periods". Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution.For this queueing model, we are able to derive closed-form expressions for the steadystate probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials.Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples.Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modelling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty. 1

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Section: Continuous-time Modelsmentioning

confidence: 99%

Analysis of a discrete-time single-server queue with an occasional extra server

Bruneel

Wittevrongel

2017

Performance Evaluation

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“…4). This is especially worth attention as we have a moderate offered load of 49% and rather simple arrival process (two modulating (27). Each curve represents a different moment in time states).…”

Section: Examplementioning

confidence: 98%

“…Queue size distributions for a buffer size equal to 50 and the MMPP parameters given in(27). Each curve represents the distribution of the queue length at a different moment in time Full buffer probability versus the buffer size for MMPP parameters given in…”

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confidence: 99%

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Transient analysis of the MMPP/G/1/K queue

Chydziñski

2006

Telecommun Syst

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“…Hence it is pertinent to develop numerical techniques to solve the resulting birth and death equations and to gain an insight into the behaviour of the various system characteristics [12]. In queueing networks theory there are only a few exact results on the transient solutions [1,[9][10][11]16].…”

Section: Introductionmentioning

confidence: 99%

Transient solution of a link with finite capacity supporting multiple streams

Parthasarathy¹,

Selvamuthu²

2001

Performance Evaluation

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Most of the real-time applications involve connection establishment in point-to-point (unicast) communication in computer networks that can be modelled as a problem of resource allocation and resource sharing. These applications have different streams, such as video, voice, graphics, etc., each having different arrival rates. A stream (call) will be admitted into the network only if the network guarantees the 'Quality of Service' (QoS) parameters of the call such as bandwidth, delay, jitter and loss probability. Otherwise, the call is rejected. Hence, the study of call blocking probabilities in such communication networks is a problem of growing importance. Application of queueing theory to solve these problems has received considerable attention in the research community.Due to the real-time nature of the applications, it is pertinent to study the time-dependent behaviour of such systems. In this paper, a real-time unicast communication has been modelled as a multi-dimensional Markov process to obtain the time-dependent blocking probabilities. The time-dependent system size probabilities, means, variances and correlation coefficients of different streams are also obtained.

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Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control

Cited by 13 publications

References 29 publications

Analysis of a discrete-time single-server queue with an occasional extra server

Analysis of a discrete-time single-server queue with an occasional extra server

Transient analysis of the MMPP/G/1/K queue

Transient solution of a link with finite capacity supporting multiple streams

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