Discrete-time multiserver queues with geometric service times

Gao, Peixia; Wittevrongel, Sabine; Bruneel, Herwig

doi:10.1016/s0305-0548(02)00155-7

Cited by 47 publications

(29 citation statements)

References 17 publications

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“…Extensions to more general service-time distributions include [23,24,25,26,27,28,29,30]. In [23,24], arbitrary-length constant service times (i.e., m-slot service times (m ≥ 1) instead of 1-slot service times) are considered.…”

Section: Discrete-time Modelsmentioning

confidence: 99%

“…An extension to a 2-state Markovian correlated arrival process is considered in [26]. Geometrically distributed service times have been similarly investigated in [27,28,29]: system content analysis in [27], relationship between the pgfs of customer delay and system content in [28], extension to correlated arrival processes in [29]. For the case of independent arrivals, [30] adds the extra complication of customers with balking behavior, i.e., if all the servers are busy, an arriving customer either enters with some probability or balks with the complementary probability.…”

Section: Discrete-time Modelsmentioning

confidence: 99%

“…in which only the unknown parameters p(0) and p(1), defined in (27), and q(i), i ≥ 1, defined in (24), remain to be determined. We note that the pgfs G 0 (z) and H 0 (z) can also be expressed in terms of the pgf F (z) of the inter-extra times, or the related pgfF (z), defined in (7) and (10), as follows:…”

Section: System Content At the Start Of Up-periods And Down-periodsmentioning

confidence: 99%

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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: Discrete-time Modelsmentioning

confidence: 99%

Section: Discrete-time Modelsmentioning

confidence: 99%

Section: System Content At the Start Of Up-periods And Down-periodsmentioning

confidence: 99%

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

Bruneel

Wittevrongel

2017

Performance Evaluation

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“…[2] and [17]. Multiserver systems with geometrically distributed service times have been considered in [7], [8], [9] and [15]. In [3] and [4], discrete-time queueing models with multiple servers and constant service times of multiple slots have been studied, but only results in connection with the system content have been derived.…”

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confidence: 99%

Calculation of output characteristics of a priority queue through a busy period analysis

Walraevens

Fiems

Wittevrongel

et al. 2009

European Journal of Operational Research

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We consider a discrete-time infinite-capacity queueing system with a general uncorrelated arrival process, constant-length service times of multiple slots, multiple servers and a first-come-first-served queueing discipline. Under the assumption that the queueing system can reach a steady state, we first establish a relationship between the steady-state probability distributions of the system content and the customer delay. Next, by means of this relationship, an explicit expression for the probability generating function of the customer delay is obtained from the known generating function of the system content, derived in previous work. In addition, several characteristics of the customer delay, namely the mean value, the variance and the tail distribution of the delay, are derived through some mathematical manipulations. The analysis is illustrated by means of some numerical examples.Editor: Artalejo

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“…[1,2]), and larger than 1 in multiserver models (e.g. [3,4]). Furthermore, service capacity is expressed in number of customers that can be served simultaneously.…”

Section: Introductionmentioning

confidence: 99%

The Discrete-Time Queue with Geometrically Distributed Service Capacities Revisited

Walraevens

Bruneel

Claeys

et al. 2013

Analytical and Stochastic Modeling Techniques and Applications

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Abstract. We analyze a discrete-time queue with variable service capacity, such that the total amount of work that can be performed during each time slot is a stochastic variable that is geometrically distributed. We study the buffer occupancy by constructing an analogous model with fixed service capacity. In contrast with classical discrete-time queueing models, however, the service times in the fixed-capacity model can take the value zero with positive probability (service times are non-negative). We study the late arrival models with immediate and delayed access, the first model being the most natural model for a system with fixed capacity and non-negative service times and the second model the more practically relevant model for the variable-capacity model.

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Discrete-time multiserver queues with geometric service times

Cited by 47 publications

References 17 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

Calculation of output characteristics of a priority queue through a busy period analysis

The Discrete-Time Queue with Geometrically Distributed Service Capacities Revisited

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