Discrete-time Queueing System Research Articles

A Controlled Discrete-Time Queueing System as a Model for the Orders of Two Competing Companies

We consider two companies that are competing for orders. Let X1(n) denote the number of orders processed by the first company at time n, and let τ(k) be the first time that X1(n)<j or X1(n)=r, given that X1(0)=k. We assume that {X1(n),n=0,1,…} is a controlled discrete-time queueing system. Each company is using some control to increase its share of orders. The aim of the first company is to maximize the expected value of τ(k), while its competitor tries to minimize this expected value. The optimal solution is obtained by making use of dynamic programming. Particular problems are solved explicitly.

A non markovian retrial queueing system

This paper studies a discrete-time queueing system in which a customer who enters in the system with a server occupied may choose to go to the retrial group or to begin its service displacing to the retrial group the customer that was in the server. If the server is idle the arriving customer begins its service immediately.Retrial and service times are general, and the only repeated customer that can go directly to the server is the one situated at the first place of the retrial group.A complete study of the model has been curried out noting specially the analysis of a recursive algorithm implemented by Theorems 2 and 3 and the study of the waiting time of a customer in the queue and in the system. Finally numerical examples are given.

Analysis of a Queue with General Service Demands and Multiple Servers with Variable Service Capacities

We present the study of a non-classical discrete-time queuing system in which the customers each request a variable amount of service, called their “service demand”, from a system with multiple servers, each of which can provide a variable amount of service, called their “service capacity”, in each time slot. The service demands are independent from customer to customer and follow a general distribution, whereas the service capacities follow a phase-type distribution and are independent from server to server and from slot to slot. Since an exact analytical analysis for this general queuing system is infeasible, we propose several approximations for the key performance characteristics in this system such as the mean system content and the mean customer delay in steady state. The accuracy of each of these approximations is compared to simulations using several numerical examples.

Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism

In the paper a finite-capacity discrete-time queueing system with geometric interarrival times and generally distributed processing times is studied. Every time when the service station becomes idle it goes for a vacation of random duration that can be treated as a power-saving mechanism. Application of a single vacation policy is one way for the system to achieve symmetry in terms of system operating costs. A system of differential equations for the transient conditional queue-size distribution is established. The solution of the corresponding system written for double probability generating functions is found using the analytical method based on a linear algebraic approach. Moreover, the representation for the probability-generating function of the stationary queue-size distribution is obtained. Numerical study illustrating theoretical results is attached as well.

Analysis of a Discrete-Time Queueing Model with Disasters

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

Delay in a 2-State Discrete-Time Queue with Stochastic State-Period Lengths and State-Dependent Server Availability and Arrivals

In this paper, we consider a discrete-time multiserver queueing system with correlation in the arrival process and in the server availability. Specifically, we are interested in the delay characteristics. The system is assumed to be in one of two different system states, and each state is characterized by its own distributions for the number of arrivals and the number of available servers in a slot. Within a state, these numbers are independent and identically distributed random variables. State changes can only occur at slot boundaries and mark the beginnings and ends of state periods. Each state has its own distribution for its period lengths, expressed in the number of slots. The stochastic process that describes the state changes introduces correlation to the system, e.g., long periods with low arrival intensity can be alternated by short periods with high arrival intensity. Using probability generating functions and the theory of the dominant singularity, we find the tail probabilities of the delay.

The Age of Information in a Discrete Time Queue: Stationary Distribution and Non-Linear Age Mean Analysis

In this work, we investigate information freshness in a status update communication system consisting of a source-destination link. Initially, we study the properties of a sample path of the age of information (AoI) process at the destination. We obtain a general formula of the stationary distribution of the AoI, under the assumption of ergodicity. We relate this result to a discrete time queueing system and provide a general expression of the generating function of AoI in relation with the system time and the peak age of information (PAoI) metric. Furthermore, we consider three different single-server system models and we obtain closed-form expressions of the generating functions and the stationary distributions of the AoI and the PAoI. The first model is a first-come-first-served (FCFS) queue, the second model is a preemptive last-come-first-served (LCFS) queue, and the last model is a bufferless system with packet dropping. We build upon these results to provide a methodology for analyzing general non-linear age functions for this type of systems, using representations of functions as power series.

A discrete-time queueing system with three different strategies

We consider a discrete-time Geo/G/1/∞ system in which a customer that finishes its first essential service may opt to abandon the system, to receive a second optional service or to go at the head of the queue in order to receive another essential service. We study the Markov chain underlying the considered queueing system and its ergodicity condition. Using a generating function approach the distribution of the number of customers in the queue and in the system as well as their respective means are given.The busy period of an auxiliary system, that will be useful to study of the customers delay, is analysed. The distributions of the sojourn time of a customer in the server, the queue and the system are provided. In order to illustrate the effect of the parameters on several performance characteristics some numerical examples are given. Finally, a section of conclusions commenting the main research contributions of this paper is presented.

Modelling and analysis of D-BMAP/D-MSP/1 queue using RG-factorization

ABSTRACT The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer’s epoch using the -factorization technique combined with the spectral expansion method to compute the stochastic matrix . For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer’s epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city.

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

In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples.

Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks

One of the most important problems of data transmission in packet networks, in particular in wireless sensor networks, are periodic overflows of buffers accumulating packets directed to a given node. In the case of a buffer overflow, all new incoming packets are lost until the overflow condition terminates. From the point of view of network optimization, it is very important to know the probabilistic nature of this phenomenon, including the probability distribution of the duration of the buffer overflow period. In this article, a mathematical model of the node of a wireless sensor network with discrete time parameter is proposed. The model is governed by a finite-buffer discrete-time queueing system with geometrically distributed interarrival times and general distribution of processing times. A system of equations for the tail cumulative distribution function of the first buffer overflow period duration conditioned by the initial state of the accumulating buffer is derived. The solution of the corresponding system written for probability generating functions is found using the analytical approach based on the idea of embedded Markov chain and linear algebra. Corresponding result for next buffer overflow periods is obtained as well. Numerical study illustrating theoretical results is attached.

Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline

We consider a discrete-time queueing system with two queues and one server. The server is allocated in each slot to the first queue with probability $$\alpha $$ and to the second queue with probability $$1-\alpha $$ . The service times are equal to one time slot. The queues have exponentially bounded, but general, arrival distributions. The mathematical description of this system leads to a single functional equation for the joint probability generating function of the stationary system contents. As the joint stochastic process of the system contents is not amenable for exact analysis, we focus on an efficient approximation of the joint probability generating function. In particular, first we prove that the partial probability generating functions, present in the functional equation, have a unique dominant pole. Secondly, we use this information to approximate these partial probability generating functions by truncating an infinite sum. The remaining finite number of unknowns are estimated from a noise perturbed linear system. We illustrate our approach by various numerical examples and verify the accuracy by means of simulation.

A Note on Approximations of Discrete-Time Queueing Systems with Their Continuous-Time Counterparts

In this paper, we discuss the performances of approximations of discrete-time queues with their continuous-time counterparts. Taking the discrete-time single-arrival Geo/G/1, batch-arrival GeoX/G/1, and tandem queues for examples, we present the absolute and relative errors resulting from such approximations and demonstrate that they can be quite vulnerable for a considerable range of parameter values. Therefore, special attention has to be paid to such approximations, and further sophisticated analysis needs to be carried out for more accurate evaluation of discrete-time queueing systems, taking their discrete-time characteristics into account.

Робастное управление дискретной системой массового обслуживания

Робастное управление дискретной системой массового обслуживания

Performance analysis of non-exhaustive wireless sensor networks based on queueing theory

An operation mechanism of wireless sensor networks with non-exhaustive service, multiple adaptive sleep and wake-up strategies is presented on the basis of the queueing theory. The operation mechanism is proposed to dominate the workload of the necessary topology maintenance and reduce the energy consumption. The wireless sensor networks are modelled as a complex discrete-time queueing system. We derive the probability generating functions of the number of the data packets, the access delay, and we obtain the expressions of the mean number of the data packets, the mean response time, the utilisation rate and the throughput, etc. The sensitivity analyses of the performance measures are presented by the numerical experiments. Furthermore, the mathematical analyses are made on the energy consumption. Nash equilibrium and the social optimisation strategies are analysed in detail on the utility by using game theory.

An exact root-free method for the expected queue length for a class of discrete-time queueing systems

For a class of discrete-time queueing systems, we present a new exact method of computing both the expectation and the distribution of the queue length. This class of systems includes the bulk-service queue and the fixed-cycle traffic-light (FCTL) queue, which is a basic model in traffic-control research and can be seen as a non-exhaustive time-limited polling system. Our method avoids finding the roots of the characteristic equation, which enhances both the reliability and the speed of the computations compared to the classical root-finding approach. We represent the queue-length expectation in an exact closed-form expression using a contour integral. We also introduce several realistic modifications of the FCTL model. For the FCTL model for a turning flow, we prove a decomposition result. This allows us to derive a bound on the difference between the bulk-service and FCTL expected queue lengths, which turns out to be small in most of the realistic cases.

A numerical method to determine the waiting time and the idle time distributions of a GI/G/1 queue when inter-arrival times and/or service times have geometric tails

We discuss a numerical method to find the distributions of the waiting time and the idle time for discrete-time queueing systems when inter-arrival or service time distributions have geometric tails. We showed by numerical experiments that our algorithm converges rapidly, and that the execution times for practical problems are negligible.

An arriving decision problem in a discrete-time queueing system

This paper discusses a discrete-time queueing system in which an arriving customer may adopt four different strategies; two of them correspond to a LCFS discipline where displacements or expulsions occur, and in the other two, the arriving customer decides to follow a FCFS discipline or to become a negative customer eliminating the customer in the server, if any. The different choices of the involved parameters make this model to enjoy a great versatility, having several special cases of interest. We carry out a thorough analysis of the system, and using a generating function approach, we derive analytical results for the stationary distributions obtaining performance measures for the number of customers in the queue and in the system. Also, recursive formulae for calculating the steady-state distributions of the queue and system size has been developed. Making use of the busy period of an auxiliary system, the sojourn times of a customer in the queue and in the system have also been obtained. Finally, some numerical examples are given.

A Theoretical Model for Analysis of Firewalls Under Bursty Traffic Flows

Firewalls are located at the front line of the network against outside threats. Performance modeling and analysis of network firewalls help to better understand their behavior and characteristics. Moreover, having an analytical model in hand helps firewall designers avoid developing multiple design alternatives and thus considerably reduce the design costs. Moreover, the network administrators can proactively identify the performance bottlenecks of the network and fix them before any malicious attack which targets the network or the firewall itself. In this paper, we propose a novel analytical approach for performance modeling and analysis of network firewalls based on a discrete-time queuing system in which the bursty nature of the incoming traffic is taken into account, where traditional queuing models such as $M/M/1$ model fails to capture peculiar characteristics of the Internet traffic. Throughput, packet loss, delay, and firewalls CPU utilization are employed as performance evaluation indicators in our proposed model. In addition, we introduce a potential DoS attack with a very low rate which can be launched against firewalls with different burstiness factors.

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

Abstract This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment, where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ⋯, n. Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server’s working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally, some examples and numerical results are presented.