The Power-Series Algorithm for Polling Systems with Time Limits

Blanc, J.P.C.

doi:10.1017/s0269964800005167

Cited by 7 publications

(4 citation statements)

References 11 publications

(39 reference statements)

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“…The power series algorithm [4,5] aims at solving the global balance equations. To this end, the state probabilities are written as a power series and via a complex computation scheme the coefficients of these series, and thus the queue-length probabilities, are obtained.…”

Section: Introductionmentioning

confidence: 99%

Time-limited polling systems with batch arrivals and phase-type service times

et al. 2011

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Abstract. In this paper, we will develop a general framework to analyze polling systems with either the autonomous-server or the time-limited service discipline. We consider Poisson batch arrivals and phase-type service times. It is known that these disciplines do not satisfy the well-known branching property in polling system. Therefore, hardly any exact results exist in the literature. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains.

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Section: Introductionmentioning

confidence: 99%

Time-limited polling systems with batch arrivals and phase-type service times

et al. 2011

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“…To circumvent this difficulty, researchers resort to numerical methods using for instance iterative solution techniques or by using a power series algorithm. The power series algorithm [2,3] aims at solving the global balance equations. To this end, the state probabilities are written as a power series and via a complex computation scheme the coefficients of these series, and thus the queuelength probabilities, are obtained.…”

Section: Introductionmentioning

confidence: 99%

Time-limited and k-limited polling systems: A matrix analytic solution

Hanbali

Haan

Boucherie

et al. 2008

Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools

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In this paper, we will develop a tool to analyze polling systems with the autonomous-server, the time-limited, and the k-limited service discipline. It is known that these disciplines do not satisfy the well-known branching property in polling system, therefore, hardly any exact result exists in the literature for them. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains. Finally, we will show that our tool works also in the case of a tandem queueing network with a single server that can serve one queue at a time.

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“…In the sequel, we choose L = 3 and L = 4; both values are compared in section 6. It can be seen that our approach is somewhat similar to the well-known power series algorithm (PSA), discussed originally in [3,17] and frequently used by Blanc in other papers (see, e.g., [4,5,6]). The difference is that in the PSA, as used in the papers just mentioned, infinite power series are used (which rises the issue of the convergence of these series) and the parameter of interest is not σ (as in our case), but the traffic intensity or the utilization factor of the queue at hand.…”

Section: Polynomial Approximation Of the System-content Pgfmentioning

confidence: 87%

“…On the other hand, if σ = 1, the system reduces to a simple discrete-time buffer without deadlines, which is stable if and only if the mean number of customers entering the system per slot, given by λ, is strictly less than 1. We now let the time parameter k go to infinity in equation (6). Assuming the system reaches a steady state, then both functions U k (·) and U k+1 (·) converge to a common limit function U (·), which denotes the pgf of the system content at the beginning of an arbitrary slot in steady state.…”

Section: Exact Analysis Of the System-content Pgfmentioning

confidence: 99%

A discrete-time queue with customers with geometric deadlines

Bruneel

Maertens

2015

Performance Evaluation

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This paper studies a discrete-time queueing system where each customer has a maximum allowed sojourn time in the system, referred to as the "deadline" of the customer. More specifically, we model the deadlines of the consecutive customers as independent and geometrically distributed random variables. 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. Service times of the customers are deterministically equal to one slot each. For this queueing model, we are able to obtain exact formulas for such quantities as the generating function and the expected value of the system content, the mean customer delay and the deadline-expiration ratio. These formulas, however, contain infinite sums and infinite products, which implies that truncations are required to actually compute numerical values. Therefore, we also derive some easy-to-evaluate approximate results for the main performance measures, based on a polynomial approximation technique. We believe this technique, in its own right, is also one of the major (methodological) contributions of the paper.Possible applications of this type of queueing model are numerous: the (variable) deadlines could model, for instance, the fact that customers may become impatient and leave the queue unserved if they have to wait too long in line, but they could also reflect the fact that the service of a customer is not useful anymore if it cannot be delivered soon enough, etc.

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The Power-Series Algorithm for Polling Systems with Time Limits

Cited by 7 publications

References 11 publications

Time-limited polling systems with batch arrivals and phase-type service times

Time-limited polling systems with batch arrivals and phase-type service times

Time-limited and k-limited polling systems: A matrix analytic solution

A discrete-time queue with customers with geometric deadlines

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