Discrete-time queues with variable service capacity: a basic model and its analysis

In this paper, we present a basic discrete-time queueing model whereby the service process is decomposed in two (variable) components: the demand of each customer, expressed in a number of work units needed to provide full service of the customer, and the capacity of the server, i.e., the number of work units that the service facility is able to perform per time unit. The model is closely related to multi-server queueing models with server interruptions, in the sense that the service facility is able to deliver more than one unit of work per time unit, and that the number of work units that can be executed per time unit is not constant over time. Although multi-server queueing models with server interruptions-to some extent-allow us to study the concept of variable capacity, these models have a major disadvantage. The models are notoriously hard to analyze and even when explicit expressions are obtained, these contain various unknown probabilities that have to be calculated numerically, which makes the expressions difficult to interpret. For the model in this paper, on the other hand, we are able to obtain explicit closed-form expressions for the main performance measures of interest. Possible applications of this type of queueing model are numerous: the variable service capacity could model variable available bandwidths in communication networks, a varying production capacity in factories, a variable number of workers in an HR-environment, varying capacity in road traffic, etc

Performance analysis of buffers with train arrivals and correlated output interruptions

In this paper, we study a discrete-time buffer system with a timecorrelated packet arrival process and one unreliable output line. In particular, packets arrive to the buffer in the form of variable-length packet trains at a fixed rate of exactly one packet per slot. The packet trains are assumed to have a geometric length, such that each packet has a fixed probability of being the last of its corresponding train. The output line is governed by a Markovian process, such that the probability that the line is available during a slot depends on the state of the underlying J-state Markov process during that slot. First, we provide a general analysis of the state of the buffer system based on a matrix generating functions approach. This also leads to an expression for the mean buffer content. Additionally, we take a closer look at the distributions of the packet delay and the train delay. In order to make matters more concrete, we next present a detailed and explicit analysis of the buffer system in case the output line is governed by a 2-state Markov process. Some numerical examples help to visualise the influence of the various model parameters

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

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 steady-state 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 modeling 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. (C) 2017 Elsevier B.V. All rights reserved