In this paper, we investigate a single-server discrete-time queueing system subject to two independent batch Bernoulli arrival processes, each supplying the queue with different customer classes. The two classes of customers have different priority levels in the queue, and different service-time distributions. The studied priority mechanism is time-limited, i.e., customers of the high-priority class cannot overtake customers of lower priority if the latter arrived at least N slots earlier than the former. The parameter N makes the mechanism versatile, spanning a bridge between absolute (fixed) priority and slot-bound priority (see De Clercq et al. in Math Probl Eng. doi: 10.1155/2012/425630 , 2012 ). The time-limited overtake priority mechanism maintains levels of fairness that are unattainable by a pure absolute priority mechanism, and offers more service differentiation than the slot-bound priority alternative studied earlier. By using a censoring argument, we obtain expressions for the steady-state probability generating functions of the delays of both customer classes, as well as the steady-state joint probability generating function of the system content, by using a censoring argument.

Exact analysis of tandem priority queues is a difficult problem. In this paper, we model the output process of the first stage as a three-state Markov chain and analyze the second stage. The arrival process of this second stage is the superposition of this output process and an uncorrelated arrival process. We calculate the joint probability generating function of the number of high- and low-priority packets in the second stage and show that two implicit functions appear in this expression. We demonstrate how to deal with these implicitly defined functions in the calculation of moments.

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn’t actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into ‘during the same time-slot’. In other words, it doesn’t specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multi-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.

The paper we present here introduces a new priority mechanism in discrete-time queueing systems. It is a milder form of priority when compared to HoL priority, but it favors customers of one type over the other when compared to regular FCFS. It also provides an answer to the starvation problem that occurs in HoL priority systems. In this new priority mechanism, customers of different priority classes entering the system during the same time slot are served in order of their respective priority class—hence the name slot-bound priority. Customers entering during different slots are served on an FCFS basis. We consider two customer classes (pertaining to two levels of priority) such that type-1 customers are served before type-2 customers that enter the system during the same slot. A general independent arrival process and generally distributed service times are assumed. Expressions for the probability generating function (PGF) of the system content (number of type- customers, ) in regime are obtained using a slot-to-slot analysis. The first moments are calculated, as well as an approximation for the probability mass functions associated with the found PGFs. Lastly, some examples allow us some deeper insight into the inner workings of the slot-bound priority mechanism.

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.

For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive priority systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this decomposition property.

In this paper we investigate a single-server discrete-time queueing system with single-slot service times. The stationary ergodic arrival process this queueing system is subject to, satisfies a regeneration property when there are no arrivals during a slot. Expressions for the mean and the variance of the queue content in steady state are obtained for this broad class which includes among others autoregressive arrival processes and M/G/infinity-input or train arrival processes. To illustrate our results, we then consider a number of numerical examples.

Recent years have seen a considerable increase of attention devoted to Poisson's equation for Markov chains, which now has attained a central place in Markov chain theory, due to the extensive list of areas where Poisson's equation pops up: perturbation analysis, Markov decision processes, limit theorems of Markov chains, etc. all find natural expression when viewed from the vantage point of Poisson's equation. We describe how the use of generating functions helps solve Poisson's equation for different types of structured Markov chains and for driving functions, and point out some applications. In particular, we solve Poisson's equation in the transform domain for skip-free Markov chains and Markov chains with linear displacement. Closed-form solutions are obtained for a class of driving functions encompassing polynomial functions and functions with finite support.

A well-known problem with priority policies is starvation of delay-tolerant traffic. Additionally, insufficient control over delay differentiation (which is needed for modern network applications) has incited the development of sophisticated scheduling disciplines. The priority policy we present here has the benefit of being open to rigorous analysis. We study a discrete time queueing system with a single server and single queue, in which N types of customers enter pertaining to different priorities. A general i.i.d. arrival process is assumed and service times are generally distributed. We divide the time axis into 'frames' of fixed size (counted as a number of time-slots), and re-order the customers that enter the system during the same frame such that the high-priority customers are served first. This paper gives an analytic approach to studying such a system, and in particular focuses on the system content (meaning the customers of each type in the system at random slot marks)in stationary regime, and the delay distribution of a random customer. Clearly, in such a system the frame's size is the key factor in the delay differentiation between the N priority classes. The numerical results at the end of this paper illustrate this observation.

For several specific discrete-time queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. This paper aims to show that this decomposition can be applied to for a large class of discrete-time queueing systems with vacations. The analysis builds on results obtained by Fuhrmann and Cooper concerning continuous-time queueing systems with Poissonian arrivals. Through some examples we show that the queueing analysis can be considerably simplified using this decomposition property.

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn't actually say what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into 'during the same time-slot'. In other words, it doesn't specify in which order such customers are served? When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect things like queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a dual-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in an arbitrary order. It will be our goal to search for queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the arbitrary service order is equivalent to FCFS if different customers have different arrival epochs and if the arrival epochs are independent of class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a bounded continuous time frame. The equivalent problem in continuous time has thus far only been solved for a Poissonian arrival process, and some specific cases of MAP.

A well-known problem with priority policies is starvation of delay-tolerant traffic. Furthermore insufficient control over delay differentiation (which is needed for modern network applications) has incited the development of other scheduling disciplines. Processor sharing is one of many solutions to this problem. The priority policy we present here has the added benefit of being more open to rigorous analysis. We study a discrete-time queueing system with a single server and single queue, in which N types of customers enter (we will refer to packets as customers) pertaining to different priorities. A general i.i.d. arrival process is assumed and service times are deterministic. We divide the time axis into 'frames' of fixed size (counted as a number of time-slots), and reorder the customers that enter the system during the same frame such that the high-priority customers are served first. This paper gives an analytic approach to studying such a system, and in particular focuses on the queue content (meaning the customers of each type in the system at random slotmarks) in stationary regime. Clearly the frame's size is the key factor in the delay differentiation between the two priority classes. The numerical results at the end of this paper illustrate this fact.