Discrete-time Queueing Research Articles

Mathematical models and algorithms for the optimal control of FIFO-queues in shared memory

The paper deals with the problems of optimal partition of shared memory for two and arbitrary number of queues in discrete time. Mathematical models and algorithms for the optimal partition of memory between queues, minimizing data loss, are presented.

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.

Analytically simple and computationally efficient solution to GI[formula omitted]/Geom/1 queues involving heavy-tailed distributions

This paper provides an analytically simple and computationally efficient solution to the discrete-time queueing model GIX /Geom/1. Using the imbedded Markov chain method, the analysis has been carried out for the late-arrival system and the results for the early-arrival system have been derived from those of the late-arrival system. The distributions of the numbers in the system at various epochs are all found in terms of roots of the underlying characteristic equation. Numerical results are also discussed.

A discrete-time queueing network approach to performance evaluation of autonomous vehicle storage and retrieval systems

In this paper, we present a method for performance evaluation of autonomous vehicle storage and retrieval systems (AVS/RSs) with tier-captive single-aisle vehicles. A discrete-time open queueing network approach is applied. The data obtained from the evaluation of the lift and vehicle movements can be used directly as input for the general discrete service time distributions of the queueing network. Furthermore, the approach allows for the computation of the retrieval transaction time distribution as well as of the distribution of the number of transactions waiting to be stored. Consequently, not only expected values and variances but also quantiles of the performance measures can be obtained. Comparison to discrete-event simulation quantifies approximation errors resulting from the decomposition approach in the discrete-time domain. Moreover, the errors obtained by the discrete-time approach are compared to the errors obtained using a continuous-time open queueing network approach. Finally, it will be outlined how the model can be used for designing AVS/RSs according to given system requirements, such as storage capacity, throughput, height and length of the system as well as the 95% quantile of the retrieval transaction time.

A two-class global FCFS discrete-time queueing model with arbitrary-length constant service times

We analyze a discrete-time queueing model where two types of customers, each having their own dedicated server, are accommodated in one single FCFS queue. Service times are deterministically equal to s≥1 time slots each. New customers enter the system according to a general independent arrival process, but the types of consecutive customers may be nonindependent. As a result, arriving customers may (or may not) have the tendency to cluster according to their types, which may lead to more (or less) blocking of one type by the opposite type. The paper reveals the impact of this blocking phenomenon on the achievable throughput, the (average) system content, the (average) customer delay and the (average) unfinished work. The paper extends the results of earlier work where either the service times were assumed to be constant and equal to 1 slot each, or the customers all belonged to the same class. Our results show that, in case of Poisson arrivals, for given traffic intensity, the system-content distribution is insensitive to the length (s) of the service times, but the (mean) delay and the (mean) unfinished work in the system are not. In case of bursty arrivals, we find that all the performance measures are affected by the length (s) of the service times, for given traffic intensity.

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.

System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy

In this paper, we deal with a discrete-time $Geo/G/1$ queueing system under the control of Min($N, V$)-policy in which the server takes single vacation whenever the system becomes empty. The Min($N, V$)-policy means that the server commences its service once the number of waiting customers reaches threshold $N$ or when its vacation time ends with at least one but less than $N$ customers waiting for processing, whichever occurs first. Otherwise, if no customer is presenting at the end of the server vacation, the server remains idle until the first arrival occurs. Under these assumptions, the $z$-transform expressions for the transient queue size distribution at time epoch $n^+$ are obtained by employing the renewal process theory and the total probability decomposition technique. Based on the transient analysis, the explicit recursive formulas of the steady-state queue length distribution at time epochs $n^+$, $n$, $n^-$ and outside observer's time epoch are derived, respectively. Additionally, the stochastic decomposition structure is presented and some other performance measures are also discussed. Furthermore, some computational experiments are implemented to demonstrate the significant application value of the recursive formulas for the steady-state queue size in designing system capacity. Finally, the optimal threshold of $N$ for economizing the system cost is numerically determined.

Dynamic control policy for the discrete-time queue

In queueing system, when the server is idle, one of control policy for determining the timing of server activated is when a fixed number N of customers accumulates for service in the system. However, on some practical situations such as watching video file online, the threshold value N may vary depending on different situations. For such situation, this paper studies a random N-policy Geo/G/1 queue, where N is determined as each new cycle begins. We derive some system characteristics, such as system size, idle period, busy period and sojourn time, by using the probability generating function and supplementary variable technique. We also prove that the system size of a continuous-time random N-policy M/G/1 queue can be approximated by the presented discrete-time model.

Dynamic control policy for the discrete-time queue

Dynamic control policy for the discrete-time queue

Optimization of Access Threshold for Cognitive Radio Networks with Prioritized Secondary Users

We propose an access control scheme in cognitive radio networks with prioritized Secondary Users (SUs). Considering the different types of data in the networks, the SU packets in the system are divided into SU1 packets with higher priority and SU2 packets with lower priority. In order to control the access of the SU2 packets (including the new arrival SU2 packets and the interrupted SU2 packets), a dynamic access threshold is set. By building a discrete-time queueing model and constructing a three-dimensional Markov chain with the number of the three types of packets in the system, we derive some performance measures of the two types of the SU packets. Then, with numerical results, we show the change trends for the different performance measures. At last, considering the tradeoff between the throughput and the average delay of the SU2 packets, we build a net benefit function to make optimization for the access threshold.

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 discrete-time queues with general service demands and finite-support service capacities

In this paper, we study a non-classical discrete-time queueing model with variable service demands and variable service capacities. Specifically, we consider the case where the service capacities of the system, i.e., the numbers of work units that the system can perform during each slot, are independent from slot to slot and have an identical general distribution with finite support. New customers enter the system according to a general independent arrival process. The service demands of the customers, i.e., the numbers of work units of service that each customer requires from the system, are general independent. We present an analytical technique to analyze the behavior of this queueing system. The analysis leads to expressions for the probability generating functions and the moments of the unfinished work in the system, the queueing delay of an arbitrary customer and the number of customers in the system in steady state. We also derive approximations for the tail probabilities of the system content and the customer delay. Numerical results are included to illustrate the impact of the various model parameters on the system performance.

Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity

This paper analyzes a single-server discrete-time queueing model with general independent arrivals, where the service process of the server is characterized in two steps. Specifically, the model assumes that (1) each customer represents a random, arbitrarily distributed, amount of work for the server, the service demand, and (2) the server disposes of a fixed number of work units that can be executed per slot, the service capacity. For this non-classical queueing model, we obtain explicit closed-form results for the probability generating functions (pgf’s) of the unfinished work in the system (expressed in work units) and the queueing delay of an arbitrary customer (expressed in time slots). Deriving the pgf of the number of customers in the system turns out to be hard, in general. Nevertheless, we derive this pgf explicitly in a number of special cases, i.e., either for geometrically distributed service demands, and/or for Bernoulli arrivals or geometric arrivals. The obtained results show that the tail distributions of the unfinished work, the customer delay and the system content all exhibit a geometric decay, with semi-analytic formulas for the decay rates available. Another interesting conclusion is that, for a given system load, the mean customer delay converges to constant limiting values when the service capacity per slot goes to infinity, and either the mean arrival rate or the mean service demand increases proportionally. Accurate approximative analytical expressions are available for these limiting values.

Analysis of Discrete-time queues with correlated arrivals, negative customers and server interruption

This paper analysis a discrete time infinite capacity queueing system with correlated arrival and negative customers served by two state Markovian server. Positive customers are generated according to the first order Markovian arrival process with geometrically distributed lengths of On periods and Off periods. Further, the geometrically distributed arrival of negative customer removes the positive customers is any, and has no effect when the system is empty. The server state is a two state Markov chain which alternate between Good and Bad states with geometrically distributed service times. Closed-form expressions for mean queue length, unfinished work and sojourn time distributions are obtained. Numerical illustrations are also presented.

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.

Relationship between randomized F-policy and randomized N-policy in discrete-time queues

This paper investigates the interrelationship between the randomized F- policy and randomized N- policy in discrete-time queues with start-up time. The F-policy queuing system deals with the issue of controlling arrivals. The N-policy queueing system deals with the issue of controlling service, that is, when all the customers are served in the queue, the server is deactivated until N customers are accumulated in the queue. Using the recursive method, the steady-state probabilities are determined for both models. The relationships between the discrete-time G e o/G e o/1/K queues with (p, F)- and (q, N)-policies are established by a series of propositions. The benefit made by interrelationship is that the solution of one queue can be deduced from the other queue readily. Various performance measures and numerical results are carried out for illustration purposes. These queues are common in wireless communication systems, production systems, manufacturing systems.

Explicit solution for the stationary distribution of a discrete-time finite buffer queue

We consider a discrete-time single server queue with finite buffer. The customers arrive according to a discrete-time batch Markovian arrival process with geometrically distributed batch sizes and the service time is one time slot. For this queueing system, we obtain an exact closed-form expression for the stationary queue length distribution. The expression is in a form of mixed matrix-geometric solution.

Modeling discrete-time analytical models based on random early detection: Exponential and linear

Congestion control is among primary topics in computer network in which random early detection (RED) method is one of its common techniques. Nevertheless, RED suffers from drawbacks in particular when its "average queue length" is set below the buffer's "minimum threshold" position which makes the router buffer quickly overflow. To deal with this issue, this paper proposes two discrete-time queue analytical models that aim to utilize an instant queue length parameter as a congestion measure. This assigns mean queue length (mql) and average queueing delay smaller values than those for RED and eventually reduces buffers overflow. A comparison between RED and the proposed analytical models was conducted to identify the model that offers better performance. The proposed models outperform the classic RED in regards to mql and average queueing delay measures when congestion exists. This work also compares one of the proposed models (RED-Linear) with another analytical model named threshold-based linear reduction of arrival rate (TLRAR). The results of the mql, average queueing delay and the probability of packet loss for TLRAR are deteriorated when heavy congestion occurs, whereas, the results of our RED-Linear were not impacted and this shows superiority of our model.

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.