Discrete-time Queueing Research Articles

A performance model for an asynchronous optical buffer

We investigate the behaviour of an asynchronous optical buffer by means of a continuous-time queuing model. Through a limit procedure, previously obtained results for a discrete-time queuing model are translated to a continuous-time setting. We also show that the same results can be obtained by a direct analysis using Laplace transforms. Closed-form expressions are obtained for the cases of exponentially distributed burst sizes, deterministic burst sizes and mixtures of deterministic burst sizes. The performance of asynchronous optical buffers shows the same characteristics as that of synchronous optical buffers: a reduction in throughput due to the creation of voids on the outgoing channel and a burst loss probability that is strongly influenced by the choice of fiber delay line granularity. The optimal value of the latter depends on the burst size distribution and the offered load.

DISCRETE-TIME QUEUE WITH VARIABLE SERVICE CAPACITY

This paper considers a discrete-time queueing system with variable service capacity. Using the supplementary variable method and the generating function technique, we compute the joint probability distribution of queue length and remaining service time at an arbitrary slot boundary, and also compute the distribution of the queue length at a departure time.

A discrete-time queueing model with periodically scheduled arrival and departure slots

We consider a time-slotted queueing model where each time slot can either be an arrival slot, in which new packets arrive, or a departure slot, in which packets are transmitted and hence depart from the queue. The slot scheduling strategy we consider describes periodically, and for a fixed number of time slots, which slots are arrival and departure slots. We consider a static and a dynamic strategy. For both strategies, we obtain expressions for the probability generating function of the steady-state queue length and the packet delay. The model is motivated by cable-access networks, which are often regulated by a request–grant procedure in which actual data transmission is preceded by a reservation procedure. Time slots can then either be used for reservation or for data transmission.

Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.

Performance analysis of a discrete-time queuing system with a correlated train arrival process

In this paper, we present an exact transient and steady-state discrete-time queuing analysis of a statistical multiplexer with a finite number of input links and whose arrival process is correlated and consists of a train of a fixed number of fixed-length packets. The functional equation describing this queuing model is manipulated and transformed into a mathematical tractable form. This allows us to derive the transient probability generating function (pgf) of the buffer occupancy. From this transient pgf, time-dependent performance measures such as transient probability of empty buffer, transient mean of buffer occupancy and instantaneous packet overflow probabilities are derived. By applying the final-value theorem, the corresponding exact expressions for the steady-state pgf of the queue length and packet arrivals are derived. We also show how the transient analysis provides insights into the derivation of the system's busy period distribution. Closed-form expressions for the mean packet and message delays are also provided. The paper presents significant results on the transient and steady-state analysis of statistical multiplexers with N input links and correlated train arrivals.

Performance analysis of discrete-time queue GI/G/1 with negative arrivals

In this paper, we consider a discrete-time GI/G/1 queueing model with negative arrivals. By deriving the probability generating function of actual service time of ordinary customers, we reduced the analysis to an equivalent discrete-time GI/G/1 queueing model without negative arrival, and obtained the probability generating function of buffer contents and random customer delay.

A queue with semiperiodic traffic

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

A queue with semiperiodic traffic

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

Mean Buffer Contents in Discrete-Time Single-Server Queues with Heterogeneous Sources

We consider discrete-time single-server queues fed by independent, heterogeneous sources with geometrically distributed idle periods. While being active, each source generates some cells depending on the state of the underlying Markov chain. We first derive a general and explicit formula for the mean buffer contents in steady state when the underlying Markov chain of each source has finite states. Next we show the applicability of the general formula to queues fed by independent sources with infinite-state underlying Markov chains and discrete phase-type active periods. We then provide explicit formulas for the mean buffer contents in queues with Markovian autoregressive sources and greedy sources. Further we study two limiting cases in general settings, one is that the lengths of active periods of each source are governed by an infinite-state absorbing Markov chain, and the other is the model obtained by the limit such that the number of sources goes to infinity under an appropriate normalizing condition. As you will see, the latter limit leads to a queue with (generalized) M/G/? input sources. We provide sufficient conditions under which the general formula is applicable to these limiting cases.

Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience

ABSTRACT This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt[14] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain. We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed.

Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained. In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon. Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems.

Delay differentiation by reserving space in queue

The delay of both high and low priority packets in a discrete-time queue operating under a reservation discipline is studied. The delay distribution is obtained from which closed-form expressions for the mean, variance and the tail distribution are derived. In an example, performance is compared with that of Absolute Priority and FIFO.

Workload Process, Waiting Times, and Sojourn Times in a Discrete Time MMAP[K]/SM[K]/1/FCFS Queue

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.

The Waiting Time Distribution of a Type k Customer in a Discrete-Time MMAP[K]/PH[K]/c (c = 1, 2) Queue Using QBDs

This paper presents an improved method to calculate the delay distribution of a type k customer in a first-come-first-serve (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements, and c servers, with c = 1, 2 (the MMAP[K]/PH[K]/c queue). The first algorithms to compute this delay distribution, using the GI/M/1 paradigm, were presented by Van Houdt and Blondia [Van Houdt, B.; Blondia, C. The delay distribution of a type k customer in a first come first served MMAP[K]/PH[K]/1 queue. J. Appl. Probab. 2002, 39 (1), 213–222; The waiting time distribution of a type k customer in a FCFS MMAP[K]/PH[K]/2 queue. Technical Report; 2002]. The two most limiting properties of these algorithms are: (i) the computation of the rate matrix R related to the GI/M/1 type Markov chain, (ii) the amount of memory needed to store the transition matrices A l and B l . In this paper we demonstrate that each of the three GI/M/1 type Markov chains used to develop the algorithms in the above articles can be reduced to a QBD with a block size which is only marginally larger than that of its corresponding GI/M/1 type Markov chain. As a result, the two major limiting factors of each of these algorithms are drastically reduced to computing the G matrix of the QBD and storing the 6 matrices that characterize the QBD. Moreover, these algorithms are easier to implement, especially for the system with c = 2 servers. We also include some numerical examples that further demonstrate the reduction in computational resources.

Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations

This paper considers a finite-buffer discrete-time Geo X / G Y /1/ K+ B queue with multiple vacations that has a wide range of applications including high-speed digital telecommunication systems and various related areas. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive a set of linear equations to compute the steady-state departure-epoch probabilities based on the embedded Markov chain technique. Next, we present numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as the moments of the number of packets in the queue at three different epochs, the mean delay in the queue of a packet, the loss probability and the probability that server is busy with computational experiences.

Closed-form solutions for key performance measures of packet multiplexers with finite capacity queue

We revisit the packet multiplexer with a discrete-time single server queue of limited capacity, and we derive a set of exact closed-form solutions for the major performance measures, such as the distributions of queue length, waiting time and number of lost packets per arriving batch. Assuming equally utilised inputs, the results refer to batch arrivals of binomial and Poisson distributed size, whereas they generally hold for systems with two inputs. The analysis is based on both the commonly used Early Arrival and Late Arrival queueing models, and we show that these models are equivalent at any capacity in terms of both waiting time and loss probability if the Late Arrival queue can hold one additional packet compared to its Early Arrival counterpart.

Batch Queues with Choice of Arrivals: Equilibrium Analysis and Experimental Study

We study both theoretically and experimentally the decisions players make in two queueing games with batch service. In both games, players are asked to independently decide when to join a discrete-time queue to receive service, or they may simply choose not to join it at all. Equilibrium solutions in pure and mixed strategies are constructed for two games where balking is prohibited and where it is allowed. They are then tested experimentally in a study that varies the game type (balking vs. no balking) and information structure (private vs. public information) in a 2X2 between-subject design. With repeated iterations of the stage game, all four experimental conditions result in aggregate, but not individual, behavior approaching mixed-strategy equilibrium play.

DELAY ANALYSIS OF A MULTIMEDIA ATM MULTIPLEXER WITH HOMOGENEOUS ARRIVALS

In this paper we present a delay analysis of an ATM multiplexer handling multimedia traffic of two classes: real time class, e.g., live audio and video and nonreal time class, e.g., file transfers. Packets arrive into the multiplexer in batches of general size and are output one at a time. The arrivals are homogeneous in the sense that either the arriving batch is all class-1 packets or all class-2 packets. The multiplexer is modeled as a priority discrete time queueing system with batch arrivals and geometric service. The homogeneity of the arrivals is reflected by the addition of an indicator random variable into the model. The analysis is comprehensive in that it covers the three main delay measures: the unfinished work, the busy period and the waiting time. For these measures, probability generating functions are derived, from which moments of arbitrary orders can be obtained. However, only the first moments are obtained in the article to produce expected values. Presented at the end is a special case when the arriving batches are of Poissonian size. Some numerical values are obtained and plotted for this case.

On a random area variable arising in discrete-time queues and compact directed percolation

A well-known discrete-time, single-server queueing system with mean arrival rate λ and mean departure rate μ is considered from the perspective of the area, A, swept out by the queue occupation process during a busy period. We determine the exact form of the tail of the distribution, Pr(A > x); in particular, we show that Pr(A > x) ~ Cx−1/4 exp(−Dx1/2) for all ρ ≠ 1, where ρ ≡ λ/μ, and expressions for C and D are given. For the critical case ρ = 1 we show that Pr(A > x) ~ C'x−1/3, with C' also given. A simple mapping, used in the derivation, establishes a connection with compact directed percolation on a square lattice. As a corollary, therefore, we are also able to specify the large-area asymptotic behaviour of this model at all points in the phase diagram. This extends previous scaling results, which are only valid close to the percolation threshold.

Analysis of State-Dependent Probabilistic Server Interruptions in Discrete-Time Queues

This letter considers a problem that arises in the context of statistical, state-dependent resource allocation in communication networks. Specifically, the problem concerns a single-server discrete-time queue with state-dependent probabilistic server interruptions. The system is analyzed for two forms of probabilistic behavior: exponential and hyperbolic forms. The analysis yields several performance metrics of interest, including average system occupancy, average delay, steady-state occupancy distribution.