Discrete-time Queueing System Research Articles

Asymptotics for the late arrivals problem

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays xi _{i}. We describe the model as a bivariate Markov chain, prove its ergodicity and study the joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution on a subset of its domain. This solution allows us to prove that the equilibrium distribution of the chain decays super-exponentially fast in the quarter plane. We exploit the latter result and discuss the numerical computation of the solution through a simple yet effective approximation scheme in a wide region of the parameters. Finally, we compare the features of this queueing model with the standard M / D / 1 system, showing that the congestion turns out to be very different when the traffic intensity is close to 1.

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.

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.

Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

This paper proposes a discrete-time multi-server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state-dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed-form expressions and develop a computational algorithm for calculating the steady-state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real-time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous-time counterparts.

Throughput-Delay Trade-off Scheduling in Multi-channel Downlink Wireless Networks

This paper designs a down-link resource scheduling algorithm for wireless cellular networks to achieve throughput-delay trade-off. The available bandwidth of the down-link is divided into some parallel sub-channels, and each sub-channel can be allocated to one potential user in every time-slot. This problem is modeled as a multi-user multi-server discrete-time queuing system with a time-varying connectivity. For this system, it is well-known that the classical MaxWeight algorithm has the optimal throughput, but its delay performance is very poor. To overcome this issue, we use the Lyapunov Optimization technique to design a throughput-utility maximizing algorithm that provides a good trade-off between the throughput and delay performance. Our approach is verified by both theoretical analysis and simulation evaluations.

A discrete-time queueing system with changes in the vacation times

Abstract This paper considers a discrete-time queueing system in which an arriving customer can decide to follow a last come first served (LCFS) service discipline or to become a negative customer that eliminates the one at service, if any. After service completion, the server can opt for a vacation time or it can remain on duty. Changes in the vacation times as well as their associated distribution are thoroughly studied. An extensive analysis of the system is carried out and, using a probability generating function approach, steady-state performance measures such as the first moments of the busy period of the queue content and of customers delay are obtained. Finally, some numerical examples to show the influence of the parameters on several performance characteristics are given.

Delay analysis of a discrete-time GI − GI − 1 queue with reservation-based priority scheduling

ABSTRACTIn this paper, we study a discrete-time single-server queueing system where the arriving traffic consists of both delay-sensitive and delay-tolerant streams. We analyze the effects of a reservation-based scheduling discipline on the delay characteristics of both types of traffic. Under this discipline, a placeholder packet is inserted in the queue to accommodate future delay-sensitive arrivals, allowing them to bypass a part of the queue, without suppressing the delay-tolerant traffic.On the basis of a probability generating functions (pgf) approach, we analyze the system state distribution and the delay distribution for either traffic type and present closed-form expressions for the expected delays as well as the tail probabilities of the packet delays. We illustrate the results comparing the delay performance of our reservation-based scheduling discipline to the performance achieved by the traditional Absolute Priority (AP) and First In First Out (FIFO) disciplines.

Analysis of a discrete-time queue with time-limited overtake priority

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.

A discrete-time queueing system with server breakdowns and changes in the repair times

We consider a discrete-time queueing system in which the arriving customers can decide to follow a LCFS discipline or to join the queue. Breakdowns can occur with geometrical lifetime and repair times governed by an arbitrary distribution. The repair times can exert changes governed by a geometrical law. We carry out a thorough study of the model deriving analytical results for the stationary distributions. We obtain generating functions of the number of customers in the queue and in the system. We also obtain the generating function of the repair times taking into account possible changes in the remaining repair times. The generating functions of the busy period, sojourn time in the server, sojourn time in the queue as well as some performance measures are also provided.

Transient analysis of a finite source discrete-time queueing system using homogeneous Markov system with state size capacities (HMS/c)

In this article, a finite source discrete-time queueing system is modeled as a discrete-time homogeneous Markov system with finite state size capacities (HMS/c) and transition priorities. This Markov system is comprised of three states. The first state of the HMS/c corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the third state which represents the system's queue. In order to examine the variability of the state sizes recursive formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence the probability mass function of each state size can be evaluated. Also the expected time in queue is computed by means of the interval transition probabilities. The theoretical results are illustrated by a numerical example.

Optimal Beamforming for Video Streaming in Multiantenna Interference Networks via Diffusion Limit

In this paper, we consider queue-aware beamforming control for video streaming applications in multiantenna interference network. Using heavy traffic approach, we derive the diffusion limit for the discrete time queuing system and obtain an ergodic excess beamforming control problem for the diffusion limit system. The ergodic control problem is to minimize the average power costs of the access points subject to the constraints on the playback interruption costs and buffer overflow costs of the mobile users. To deal with the queue coupling challenge, we utilize the weak interference coupling property in the network. Using the theories of calculus and perturbation techniques, we derive a closed-form approximate priority function of the optimality equation and the associated error bound. Based on this approximation, we propose a low complexity queue-aware beamforming control algorithm, which is asymptotically optimal for sufficiently small cross-channel path gain. Finally, the proposed scheme is compared with various baselines through simulations and it is shown that significant performance gain can be achieved.

A discrete-time queue with customers with geometric deadlines

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.

A discrete-time system with service control and repairs

Abstract This paper discusses a discrete-time queueing system with starting failures in which an arriving customer follows three different strategies. Two of them correspond to the LCFS (Last Come First Served) discipline, in which displacements or expulsions of customers occur. The third strategy acts as a signal, that is, it becomes a negative customer. Also examined is the possibility of failures at each service commencement epoch. We carry out a thorough study of the model, deriving analytical results for the stationary distribution. We obtain the generating functions of the number of customers in the queue and in the system. The generating functions of the busy period as well as the sojourn times of a customer at the server, in the queue and in the system, are also provided. We present the main performance measures of the model. The versatility of this model allows us to mention several special cases of interest. Finally, we prove the convergence to the continuous-time counterpart and give some numerical results that show the behavior of some performance measures with respect to the most significant parameters of the system

Comments on: Queueing models for the analysis of communication systems

The paper under discussion is a well-written exposition on the performance modeling of communication systems by discrete-time queueing systems, and their analysis. It basically consists of two parts: a review of the literature, focusing on the modelling of information streams and on scheduling disciplines (Sects. 2, 3), and a demonstration of some key methods for the analysis of discrete-time queueing systems, focusing on a particular two-class discrete-time queue with correlated arrivals and two priority classes (Sects. 4–6). In Sect. 1 of the present note, we make some introductory comments. In Sect. 2, realizing that the literature review in Bruneel et al. (TOP, 2014) is authoritative and extensive, we focus on a few adjacent topics which fall outside the scope of Bruneel et al. (TOP, 2014) but which in our view may also be of some interest. Finally, in Sect. 3, we discuss the analysis in Sects. 4–6 of Bruneel et al. (TOP, 2014).

A Discrete-Time Queueing Model with Service Upgrade

In this paper we analyze a discrete-time queueing system where the server decides whento upgrade the service depending on the information carried by the incoming message. We carry outan extensive analysis of the system developing recursive formulae and generating functions for thestationary distribution of the number of customers in the queue, the system, the busy period and thesojourntimeas well as some numerical examples.

The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times

In this paper, we analyse the behaviour of a discrete-time single-server queueing system with general service times, equipped with the $NT$-policy. This is a threshold policy designed to reduce the number of service unit activation/deactivation cycles, whilst ensuring an acceptable delay trade-off. Once the server is deactivated, reactivation will be postponed until either $N$ customers have accumulated in the queue or the first customer has been in the queue for $T$ slots, whichever happens first. Due to this modus operandi, the system circulates between three phases: empty, accumulating and serving. We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.

Sample-path analysis of general arrival queueing systems with constant amount of work for all customers

We consider a discrete-time queueing system where the arrival process is general and each arriving customer brings in a constant amount of work which is processed at a deterministic rate. We carry out a sample-path analysis to derive an exact relation between the set of system size values and the set of waiting time values over a busy period of a given sample path. This sample-path relation is then applied to a discrete-time $$G/D/c$$ queue with constant service times of one slot, yielding a sample-path version of the steady-state distributional relation between system size and waiting time as derived earlier in the literature. The sample-path analysis of the discrete-time system is further extended to the continuous-time counterpart, resulting in a similar sample-path relation in continuous time.

Max Weight Learning Algorithms for Scheduling in Unknown Environments

We consider a discrete time queueing system where a controller makes a 2-stage decision every slot. The decision at the first stage reveals a hidden source of randomness with a control-dependent (but unknown) probability distribution. The decision at the second stage generates an attribute vector that depends on this revealed randomness. The goal is to stabilize all queues and optimize a utility function of time average attributes, subject to an additional set of time average constraints. This setting fits a wide class of stochastic optimization problems, including multi-user wireless scheduling with dynamic channel measurement decisions, and wireless multi-hop routing with multi-receiver diversity and opportunistic routing decisions. We develop a simple max-weight algorithm that learns efficient behavior by averaging functionals of previous outcomes.

Analysis of discrete-time buffers with heterogeneous session-based arrivals and general session lengths

In this paper we study the effect of session-based arrival streams on the behavior of a discrete-time queueing system with infinite storage capacity and a single output line. These session-based arrival streams are a new and realistic approach for modelling traffic generated by users in a telecommunication network. Users from an infinite population can start and end sessions during which they are active and send packets to the queueing system. Each active user generates a random, yet strictly positive number of packets per time slot. There are T different sessions types, each characterized by their own general session-length, session-incidence and bandwidth distribution.The corresponding discrete-time queueing model is analyzed by a generating-functions approach with infinite-dimensional state description. We derive the steady-state probability generating functions of the buffer content and the packet delay. From these, we obtain the mean buffer content as well as the mean packet delay and even the mean session delay. Finally, some numerical examples illustrate the influence of various parameters on the buffer behavior.

The mean queue content of discrete-time queues with zero-regenerative arrivals

This letter investigates a single-server discrete-time queuing system with single-slot service times. For a broad class of arrival processes, a closed-form expression for the mean queue content in steady state is obtained. Apart from being stationary-ergodic, the arrival process adheres to a regeneration property when there are no arrivals in a slot. Well-studied arrival processes such as autoregressive arrival processes and M/G/∞-input or train arrival processes adhere to this property.