Discrete-time Queueing Research Articles

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.

DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

Equilibrium Balking Strategies in the Geo/Geo/1 Queues with Server Breakdowns and Repairs

HeThis paper studies the equilibrium behavior of customers in the Geo/Geo/1 queue with server breakdowns and repairs. The reliability factor is introduced into the economic analysis and customers’ behavior analysis of the discrete-time queues. The server is subject to breakdowns and repairs when providing service to customers. Upon arrival, the customers decide for themselves whether to enter the system or balk based on a natural reward-cost structure and available information on hand. The behavior of customers is investigated with various levels of information regarding three states of server and the system queue length. By deriving and solving a set of system difference equations, we obtain the stationary distribution of the system and the mean sojourn time of an arriving customer. Equilibrium strategies for the customers under different levels of information are derived and the behaviors of the customers under these strategies are investigated. Finally we illustrate the effect of several parameters on the equilibrium behavior via numerical experiments.

Optimal design of a load balancing process using a discrete-time two-phase queueing model

Optimal design of a load balancing process using a discrete-time two-phase queueing model

Analysis of discrete-time multi-server queue with balking

This paper analyses a discrete-time multi-server queue with balking behavior. The inter-arrival and service times are assumed to be independent and geometrically distributed. If all the servers are busy, an arriving customer either enters with probability b or balks with probability 1 − b. Otherwise, the service of the arriving customer commences immediately. The system is analysed under the assumptions of an early arrival system and a late arrival system with delayed access. Closed-form expressions for the steady-state probabilities at arbitrary epochs and an outside observer's observation epoch probabilities are obtained. Various performance measures along with some numerical results in the form of a table and graphs are presented. Some special cases of the model have also been discussed. We have shown that the obtained results converge to the results of the continuous-time counterparts in the limiting case.

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.

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.

Discrete-time queues with variable service capacity: a basic model and its analysis

In this paper, we present a basic discrete-time queueing model whereby the service process is decomposed in two (variable) components: the demand of each customer, expressed in a number of work units needed to provide full service of the customer, and the capacity of the server, i.e., the number of work units that the service facility is able to perform per time unit. The model is closely related to multi-server queueing models with server interruptions, in the sense that the service facility is able to deliver more than one unit of work per time unit, and that the number of work units that can be executed per time unit is not constant over time. Although multi-server queueing models with server interruptions—to some extent—allow us to study the concept of variable capacity, these models have a major disadvantage. The models are notoriously hard to analyze and even when explicit expressions are obtained, these contain various unknown probabilities that have to be calculated numerically, which makes the expressions difficult to interpret. For the model in this paper, on the other hand, we are able to obtain explicit closed-form expressions for the main performance measures of interest. Possible applications of this type of queueing model are numerous: the variable service capacity could model variable available bandwidths in communication networks, a varying production capacity in factories, a variable number of workers in an HR-environment, varying capacity in road traffic, etc.

Batch service queue with change over times and Bernoulli schedule vacation interruption

In this paper, we model a discrete-time renewal input queue with change over times and Bernoulli schedule vacation interruption under batch service (a, c, b) policy. The service, working vacation and change over times are geometrically distributed. At the instants of a service completion, the vacation is interrupted and the server is resumed to a regular busy period with probability 1 − q if there are c or more customers in the system, or continues the vacation with probability q. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at various epochs. Some performance measures and a cost model have been presented and an optimum service rate has been obtained using geneticalgorithm. Numerical results showing the effect of the parameters of the modelon the key performance measures are presented.

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.

Discrete-time renewal input bulk service queue with changeover time

In this paper, we model a discrete-time infinite buffer renewal input single server queue with changeover time under (a, c, b) policy. The service and changeover time are geometrically distributed. The server begins service if there are at least c customers in the queue and the services are performed in batches of minimum size a and maximum size b (a ≤ c ≤ b). At the instant of service completion, if the queue size is less than c but not less than a secondary limit a, the server continues to serve and it will be in the changeover period if the queue size is a − 1. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at pre-arrival and arbitrary epochs. Based on the queue length distributions, some performance measures and cost functions have been discussed. Using a genetic algorithm, the optimum value of the service rate at minimum cost has been evaluated. Numerical results showing the effect of model parameters on the key performance measures are presented. Finally, we have derived the steady state queue length distributions of the continuous-time counterpart of our model.

A single-server discrete-time queue with correlated positive and negative customer arrivals

An MMBP/Geo/1 queue with correlated positive and negative customer arrivals is studied. In the infinite-capacity queueing system, positive customers and negative customers are generated by a Bernoulli bursty source with two correlated geometrically distributed periods. I.e., positive and negative customers arrive to the system according to two different geometrical arrival processes. Under the late arrival scheme (LAS), two removal disciplines caused by negative customers are investigated in the paper. In individual removal scheme, a negative customer removes a positive customer in service if any, while in disaster model, a negative customer removes all positive customers in the system if any. The negative customer arrival has no effect on the system if it finds the system empty. We analyze the Markov chains underlying the queueing systems and evaluate the performance of two systems based on generating functions technique. Some explicit solutions of the system, such as the average buffer content and the stationary probabilities are obtained. Finally, the effect of several parameters on the system performance is shown numerically.

Discrete-Time Geo x /G/1 Queue with Geometrically Working Vacations and Vacation Interruption

This paper treats a discrete-time batch arrival queue with geometrically working vacations and vacation interruption. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive the probability generating function (PGF) of the stationary system length at the departure epochs based on the embedded Markov chain technique. Next, we present the stationary distributions of the system lengths as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method we give a distribution for the number of the customers at the beginning of a busy period and give a stochastic decomposition formula for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

Analysis of a discrete-time queue with server subject to vacations and breakdowns

This paper studies a Geo/G/1 queue in which the server operates multiple vacations and may break down while working. After all messages have been served in the system, the server leaves for a vacation. Upon returning from the vacation, the server activates immediately to serve if there are some messages waiting in the queue. Otherwise, the server takes another vacation. This queueing system has potential application in the inbound email contact center. By using the generating function technique, we derive some important system characteristics, such as system size distributions of server on vacation/busy/broken-down, the expected lengths of server vacation period and complete period, and message's waiting time in the queue. In particular, a cost model is developed to determine the optimal parameter values of different vacation distributions that minimize the cost, and some numerical examples are presented for illustrative purpose.

Reliability indices of discrete-time Geox/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations

This paper considers the discrete-time GeoX/G/1 queueing model with unreliable service station and multiple adaptive delayed vacations from the perspective of reliability research. Following problems will be discussed: 1) The probability that the server is in a “generalized busy period” at time n; 2) The probability that the service station is in failure at time n, i.e., the transient unavailability of the service station, and the steady state unavailability of the service station; 3) The expected number of service station failures during the time interval (0, n], and the steady state failure frequency of the service station; 4) The expected number of service station breakdowns in a server’s “generalized busy period”. Finally, the authors demonstrate that some common discrete-time queueing models with unreliable service station are special cases of the model discussed in this paper.

Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism

We deduce approximations for the tail probabilities of the customer delay in a discrete-time queueing model with batch arrivals and batch service. As in telecommunications systems transmission times are dependent on packet sizes, we consider a general dependency between the service time of a batch and the number of customers within it. The model also incorporates a timer mechanism to avoid excessive delays stemming from the requirement that a service can only be initiated when the number of present customers reaches or exceeds a service threshold. The service discipline is first-come, first-served (FCFS). We demonstrate in detail that our approximations are very useful for the purpose of assessing the order of magnitude of the tail probabilities of the customer delay, except in some special cases that we discuss extensively. We also illustrate that neglecting batch-size dependent service times or a timer mechanism can lead to a devastating assessment of the tail probabilities of the customer delay, which highlights the necessity to include these features in the model. The results from this paper can, for instance, be applied to assess the quality of service (QoS) of Voice over IP (VoIP) conversations, which is typically expressed in terms of the order of magnitude of the probability of packet loss due to excessive delays.

A note on the distributional Little’s law for discrete-time queues with D-MAP arrivals and its application

For a class of discrete-time FIFO queueing systems with D-MAP (discrete-time Markovian arrival process), we present a distributional Little’s law that relates the distribution of the stationary number of customers in system (queue) with that of the stationary number of slots a customer spends in system (queue). Taking the multi-server D-MAP/D/c queue as an example, we demonstrate how this relation can be utilized to get the desired distribution of the number of customers.Sample numerical results are presented at the end.

A two-class discrete-time queueing model with two dedicated servers and global FCFS service discipline

This paper considers a simple discrete-time queueing model with two types (classes) of customers (types 1 and 2) each having their own dedicated server (servers A and B resp.). New customers enter the system according to a general independent arrival process, i.e., the total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. Service times are deterministically equal to 1 slot each. The system uses a “global FCFS” service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. As a consequence of the “global FCFS” rule, customers of one type may be blocked by customers of the other type, in that they may be unable to reach their dedicated server even at times when this server is idle, i.e., the system is basically non-workconserving. One major aim of the paper is to estimate the negative impact of this phenomenon on the queueing performance of the system, in terms of the achievable throughput, the system occupancy, the idle probability of each server and the delay. As it is clear that customers of different types hinder each other more as they tend to arrive in the system more clustered according to class, the degree of “class clustering” in the arrival process is explicitly modeled in the paper and its very direct impact on the performance measures is revealed. The motivation of our work are systems where this kind of blocking is encountered, such as input-queueing network switches or road splits.

Analysis of discrete-time queues with batch renewal input and multiple vacations

This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for partial-batch rejection policy. The blocking probability of the first-, an arbitraryand the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.

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.