Discrete-time Queueing Research Articles

On BASTA for Discrete-Time Queues

We address the issue of Bernoulli/Geometric arrivals see time-averages, BASTA, in discrete-time queues when using a variety of scheduling rules. It is well-known that BASTA/ASTA holds in the framework of a discrete-time process with an imbedded arrival process without regard to scheduling rules. However, when scheduling rules are invoked, it is not always clear whether arrivals see time averages as asserted by the general theory. We resolve this issue so that BASTA/ASTA holds regardless of the scheduling rule at play.

Numerical Computations and Tail Asymptotics for Stationary Indices of a Discrete-time Queue

Numerical Computations and Tail Asymptotics for Stationary Indices of a Discrete-time Queue

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 dynamic channel aggregation strategy with a maximum aggregation number in cognitive radio networks based on channel classification

In traditional cognitive radio networks (CRNs), users are generally divided into two categories: primary users (PUs) and secondary users (SUs). The transmission rates of different channels may vary due to the complexities of network transmissions. Therefore, in this paper, a channel classification mechanism that divides the channels in the system into high-speed channels and low-speed channels according to their transmission capacity is proposed. In addition, to improve the performance of users, a dynamic channel aggregation strategy for CRNs with two types of channels is proposed. In this strategy, packets first consider accessing high-speed channels. We set a fixed channel aggregation strategy for PU packets and a dynamic channel aggregation strategy for SU packets. The number of channel aggregation of SU packets is restricted by the maximum aggregation number. Based on the proposed strategy, a multi-service discrete-time queueing model is established and analyzed, and the performance index expressions of PU packets and SU packets are derived. Moreover, we conduct numerical experiments, obtain the system performance figures and analyze the system performance. Finally, the performance of the proposed channel aggregation strategy is compared with that of the non-aggregation strategy, verifying the effectiveness of the proposed channel aggregation strategy.

A Discrete-Time Queueing Model of a Bottleneck with an Energy-Saving Mechanism Based on Setup and Shutdown Times

Producers are encouraged to reduce their energy consumption of manufacturing systems by applying less-energy-intensive modern technologies and advanced machine tools and operating methods at the system level. In the paper, organizational and analytical solutions are combined to model the sustainable production system. Managers can study the behavior of a production system organized using energy-saving rules by changing key parameters of the input model (arrival intensity, bottleneck service rate, buffer size, setup and shutdown time) to analyze the queue size of the production system and therefore performance. A discrete-time queueing model of a single-bottleneck production line with a finite input buffer capacity is proposed. Jobs occur according to a binomial process and are processed individually, one by one, according to the natural FIFO service discipline, with a general discrete-type cumulative distribution function. The total number of jobs present in the system is bounded by a non-random fixed value N. Every time the system becomes empty, an energy-saving mechanism is started: the processing machine (server) is turned off during a geometrically distributed shutdown time. Similarly, the first job arriving into the empty system initializes a geometrically distributed setup time. Identifying renewal moments in the evolution of the model, a system of difference equations is built for the transient queue-size distribution conditioned by the state of the system at the opening. The solution is obtained explicitly in terms of probability-generating functions. In addition, the Drum-Buffer-Rope concept is proposed to reduce the energy consumption of the production line. The throughput of the production system is maximized by adjusting the time between the order arrivals and the size of the input buffer to the capacity of the bottleneck. Turning off a machine under certain conditions and slowing down non-critical machines are strategies to reduce energy consumption. A detailed illustrating numerical and simulation study of the considered model is attached as well, in which the sensitivity of the queue-size behavior to changes of the key input model parameters is investigated.

A Review of Birth-Death and Other Markovian Discrete-Time Queues

In this review article, we consider discrete-time birth-death processes and their applications to discrete-time queues. To make the analysis simpler to follow, we focus on transform-free methods and consider instances of non-birth-death Markovian discrete-time systems. We present a number of results within one discrete-time framework that parallels the treatment of continuous time models. This approach has two advantages; first, it unifies the treatment of several discrete-time models in one framework, and second, it parallels to the extent possible the treatment of continuous time models. This allows us to draw parallels and contrasts between the discrete and continuous time queues. Specifically, we focus on birth-death applications to the single server discrete-time model with Bernoulli arrivals and geometric service times and provide the reader with a simple rigorous detailed analysis that covers all five scheduling rules considered in the literature, with attention to stationary distributions at slot edges, slot centers, and prearrival epochs. We also cover the waiting time distributions. Moreover, we cover three Markovian models that fit the global balance equations. Our approach provides interesting insights into the behavior of discrete-time queues. The article is intended for those who are familiar with queueing theory basics and would like a simple, yet rigorous introductory treatment to discrete-time queues.

The Price of Anarchy of Strategic Queuing Systems

Bounding the price of anarchy, which quantifies the damage to social welfare due to selfish behavior of the participants, has been an important area of research in algorithmic game theory. Classical work on such bounds in repeated games makes the strong assumption that the subsequent rounds of the repeated games are independent beyond any influence on play from past history. This work studies such bounds in environments that themselves change due to the actions of the agents. Concretely, we consider this problem in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet at each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies. We analyze this queuing system from multiple perspectives. As a baseline measure, we first establish precise conditions on the queuing arrival rates and service capacities that ensure all packets clear efficiently under centralized coordination. We then show that if queues strategically choose servers according to independent and stationary distributions, the system remains stable provided it would be stable under coordination with arrival rates scaled up by a factor of just \(\frac{e}{e-1}\) . Finally, we extend these results to no-regret learning dynamics: if queues use learning algorithms satisfying the no-regret property to choose servers, then the requisite factor increases to 2, and both of these bounds are tight. Both of these results require new probabilistic techniques compared to the classical price of anarchy literature and show that in such settings, no-regret learning can exhibit efficiency loss due to myopia.

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 Discrete-Time Queues with Branching Arrivals

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply π-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes.

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.

Analysis of a Discrete-time Queue with Modified Batch Service Policy and Batch-size-dependent Service

Analysis of a Discrete-time Queue with Modified Batch Service Policy and Batch-size-dependent Service

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.

Modelling and optimization for cognitive radio networks with preemption backoff mechanism

In cognitive radio networks (CRNs), secondary users (SUs) are permitted to use the available spectrum in opportunity ways to optimize spectrum utilization. As legal spectrum owners, primary users (PUs) can always interrupt SUs’ transmissions when they have transmission needs. To increase SUs’ transmission continuity while guaranteeing PUs’ transmission quality, we propose a preemption backoff mechanism (PBM) with backoff threshold m and backoff parameter f. Under the centralized spectrum sharing structure, we construct a discrete-time queueing model by developing a 3-dimensional Markov-chain (3DMC). Through the model analysis, the system’s steady-state probability distribution is obtained. To assess the comprehensive performance of our proposed mechanism, we deduce various important expressions of performance indices and get the corresponding performance figures. At last, we design a cost function by compromising various performance indices and find the optimum parameter combination under the lowest system cost through the Transient Search Optimization (TSO) algorithm. The numerical and simulation results illustrate our proposed PBM can improve SUs’ transmission continuity. For example, when m=1 and f=0.05, under the parameter settings given by the experiments, the proposed PBM can reduce SU packets’ outage rate and improve throughput by about 60% at most.

First exit time for a discrete-time parallel queue

First exit time for a discrete-time parallel queue

Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service

Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service

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.

Sojourn Times in a Queueing System with Breakdowns and General Retrial Times

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue

This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a heavy-traffic limit approach. The correlation coefficient of the queue-contents is computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a hyperbola. In most server-sharing models, it is assumed that the system is work-conserving in the sense that if one of the queues is empty, a customer of the other queue is served with probability 1. In our second model, we omit this work-conserving rule such that the server can be idle in case of a non-empty queue. Contrary to what we would expect, the resulting heavy-traffic approximations reveal that both models remain different for critically loaded queues.

A Multichannel Allocation Strategy Based on Preemption Threshold and Preemption Probability in Cognitive Radio Networks

In traditional multichannel cognitive radio networks (CRNs), users are split into two different priorities. Because of the low priority of secondary users (SUs), SU packets’ transmissions are easily interrupted by primary users (PUs). In this paper, two control parameters, called preemption threshold H and preemption probability q, are used to regulate the preemption behavior of PU packets to improve the transmission performance of SU packets. When all channels in the system are occupied, the preemption behavior of PU packets will be adjusted according to the amount of SU packets that are transmitting in the system. If the amount is larger than H, the recently arrived PU packet either preempts a channel with probability q or leaves the system with probability 1 − q . The central controller manages the system’s channel usage right and determines a series of access behaviors of user packets. Considering the possible imperfect sensing, a discrete-time queueing model is developed with the proposed preemption control mechanism. Then we obtain some performance index expressions of PU and SU packets founded on the system’s state transition matrix and make the corresponding performance figures through numerical experiment. Finally, we construct a system utility function and determine the optimal preemption threshold and preemption probability through the seagull optimization algorithm (SOA). Experimental data show that the proposed mechanism by setting preemption threshold and preemption probability can significantly reduce SU packets’ outage rate and improve SU packets’ throughput rate.

Architecture design and performance analysis of a novel memory system for high-bandwidth onboard switching fabric

The development of the space–air–ground integration has put higher demands on satellite network. It needs to support high-bandwidth, large-capacity and various quality of service guarantees. However, the satellite onboard switching is facing the problem of resource constraints and special requirements of hardware complexity and scheduling efficiency. For this, this paper proposes a novel memory architecture for high-bandwidth onboard switching fabric. To reduce hardware complexity, multiplexer is used in the input side to merge k input ports into one internal bus, while the output demultiplexer splits an internal bus into k output links. Shared memory architecture is adopted in the input module to improve the memory efficiency, and a buffered crossbar is used to interconnect the input modules to the output modules. A discrete-time queuing model is formulated, and an iterative approach is used to quantitatively analyze the performance of the proposed architecture. The throughput and delay are calculated under different switch size, input load, and buffer capacity. The numerical results can serve as a guidance on deciding the required buffer size and appropriate speedup ratio. This work is validated by both simulations and FPGA implementations. Synthesize result show that using the state-of-the-art Xilinx VU13P FPGA, a switching fabric with 48 ports can be implemented and the peak throughput of the proposed architecture can reach 480 Gbps. Compared with existing combined input-crosspoint queuing, the proposed architecture can significantly reduce the packet delay and the memory resource cost.