A multi-class discrete-time queueing system under the FCFS service discipline

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.     

A semidefinite optimization approach to the steady-state analysis of queueing systems

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.     

Delay and partial system contents for a discrete-time G-D-c queue

We consider a discrete-time multiserver queueing system with infinite buffer size, constant service times of multiple slots and a first-come-first-served queueing discipline. A relationship between the probability distributions of the partial system contents and the packet delay is established. The relationship is general in the sense that it doesn’t require knowledge of the exact nature of the arrival process. By means of the relationship, results for the distribution of the partial system contents for a wide variety of discrete-time queueing models can be transformed into corresponding results for the delay distribution. As a result, a separate full analysis of the packet delay becomes unnecessary.      

Effect of global FCFS and relative load distribution in two-class queues with dedicated servers

In this paper, we investigate multi-class multi-server queueing systems with global FCFS policy, i.e., where customers requiring different types of service—provided by distinct servers—are accommodated in one common FCFS queue. In such scenarios, customers of one class (i.e., requiring a given type of service) may be hindered by customers of other classes. The purpose of this paper is twofold: to gain (qualitative and quantitative) insight into the impact of (i) the global FCFS policy and (ii) the relative distribution of the load amongst the customer classes, on the system performance. We therefore develop and analyze an appropriate discrete-time queueing model with general independent arrivals, two (independent) customer classes and two class-specific servers. We study the stability of the system and derive the system-content distribution at random slot boundaries; we also obtain mean values of the system content and the customer delay, both globally and for each class individually. We then extensively compare these results with those obtained for an analogous system without global FCFS policy (i.e., with individual queues for the two servers). We demonstrate that global FCFS, as well as the relative distribution of the load over the two customer classes, may have a major impact on the system performance.     

Queueing models for the analysis of communication systems

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.     

Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation—i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.     

Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model

This is a companion paper to Li and Zhao (Queueing Syst. 63:355–381, 2009) recently published in Queueing Systems, in which the classical preemptive priority queueing system was considered. In the current paper we consider the classical non-preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through an analysis of generating functions. This is done for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and also for the distribution of the total number of customers in the system, respectively. This complete characterization is supplemental to the existing literature, which would be useful to researchers.     

Many-Sources Delay Asymptotics with Applications to Priority Queues

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.     

Discrete-time queues with discretionary priorities

In this paper, we consider a discrete-time two-class discretionary priority queueing model with generally distributed service times and per slot i.i.d. structured inputs in which preemptions are allowed only when the elapsed service time of a lower-class customer being served does not exceed a certain threshold. As the preemption mode of the discretionary priority discipline, we consider the Preemptive Resume, Preemptive Repeat Different, and Preemptive Repeat Identical modes. We derive the Probability Generating Functions (PGFs) and first moments of queue lengths of each class in this model for all the three preemption modes in a unified manner. The obtained results include all the previous works on discrete-time priority queueing models with general service times and structured inputs as their special cases. A numerical example shows that, using the discretionary priority discipline, we can more subtly adjust the system performances than is possible using either the pure non-preemptive or the preemptive priority disciplines.     

Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1−β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z ₁,z ₂) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z ₁,z ₂) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.     

Analysis of a two-class single-server discrete-time FCFS queue: the effect of interclass correlation

In this paper, we study a discrete-time queueing system with one server and two classes of customers. Customers enter the system according to a general independent arrival process. The classes of consecutive customers, however, are correlated in a Markovian way. The system uses a “global FCFS” service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their classes. The service-time distribution of the customers is general but class-dependent, and therefore, the exact order in which the customers of both classes succeed each other in the arrival stream is important, which is reflected by the complexity of the system content and waiting time analysis presented in this paper. In particular, a detailed waiting time analysis of this kind of multi-class system has not yet been published, and is considered to be one of the main novelties by the authors. In addition to that, a major aim of the paper is to estimate the impact of interclass correlation in the arrival stream on the total number of customers in the system, and the customer delay. The results reveal that the system can exhibit two different classes of stochastic equilibrium: a “strong” equilibrium where both customer classes give rise to stable behavior individually, and a “compensated” equilibrium where one customer type creates overload.     

Flow time distributions in aK classM/G/1 priority feedback queue

In this paper, aK classM/G/1 queueing system with feedback is examined. Each arrival requires at least one, and possibly up toK service phases. A customer is said to be in classk if it is waiting for or receiving itskth phase of service. When a customer finishes its phasek ≤K service, it either leaves the system with probabilityp _k, or it instantaneously reenters the system as a classk + 1 customer with probability (1 −p _k). It is assumed thatp _k = 1. Service is non-preemptive and FCFS within a specified priority ordering of the customer classes. Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,…,k;k ≤K.     

System delay versus system content for discrete-time queueing systems subject to server interruptions

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution—actually an infinite family of quasi-stationary distributions— if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step (killing) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth–death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.      

A queueing system with linear repeated attempts,bernoulli schedule and feedback

We analyse a single-server retrial queueing system with infinite buffer, Poisson arrivals, general distribution of service time and linear retrial policy. If an arriving customer finds the server occupied, he joins with probabilityp a retrial group (called orbit) and with complementary probabilityq a priority queue in order to be served. After the customer is served completely, he will decide either to return to the priority queue for another service with probability ϑ or to leave the system forever with probability =1−ϑ, where 0≤ϑ<1. We study the ergodicity of the embedded Markov chain, its stationary distribution function and the joint generating function of the number of customers in both groups in the steady-state regime. Moreover, we obtain the generating function of system size distribution, which generalizes the well-knownPollaczek-Khinchin formula. Also we obtain a stochastic decomposition law for our queueing system and as an application we study the asymptotic behaviour under high rate of retrials. The results agree with known special cases. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.     

Performance analysis of a GI-Geo-1 buffer with a preemptive resume priority scheduling discipline

In this paper, we analyze a discrete-time GI-Geo-1 preemptive resume priority queue. We consider two classes of packets which have to be served, where one class has preemptive resume priority over the other. We show that the use of generating functions is beneficial for analyzing the system contents and packet delay of both classes. Moments and (approximate) tail probabilities of system contents and packet delay are calculated. The influence of the priority scheduling is shown by some numerical examples.     

A discrete-time Geo/G/1 retrial queue with preferred and impatient customers

This paper is concerned with a discrete-time Geo/G/1 retrial queue with preferred, impatient customers and general retrial times. We analyze the Markov chain underlying the considered queueing system and derive its ergodicity condition. The system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of priority and impatience on several performance characteristics of the system.     

A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures

This paper studies a discrete-time Geo/G/1 retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics. This work is supported by the DGINV through the project BFM2002-02189.