Batch Poisson Arrivals Research Articles

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

Performance analysis for Bernoulli feedback queues subject to disasters: a system with batch Poisson arrivals under a multiple vacation policy

ABSTRACT We analyze an M/G/1 system with batch Poisson arrivals and instantaneous Bernoulli feedback, operating under a Multiple Vacation Policy. The system is subject to disasters that occur according to an independent Poisson process and are followed by (random) repair periods with general distribution. The analysis is carried out using the supplementary variable method. The Laplace transform of the time between two consecutive disasters is obtained and the existence of the stationary regime for the system is shown. Besides obtaining the stationary distribution for the number of customers in the system, we use the information regarding the rates of occurrence of various events provided by the supplementary variable solution to obtain a great variety of additional results. These include the Laplace transform of the busy period distribution and the probability that a customer completes service. We indicate areas of application of our model to real-life systems, in particular in Vehicle ad hoc Networks (VANETs), and we use the analytic results obtained to optimize such a system under a Quality of Service constraint. Finally, we analyze a variant of the system subject to disasters even when the server is not busy.

Multi-queue system with finite shared buffer under batch Poisson and/or interrupted batch Poisson arrivals: a computational analysis

This paper studies a model of a multi-queue system with a finite common buffer shared by queues with batch arrivals. The study is performed for a few buffer-sharing policies. For batch Poisson arrivals with an arbitrary batch size distribution, the analysis benefits from the fact that the stationary state distribution has a product form. For the special case of geometrically distributed batches, closed form solutions for normalisation constant are obtained. They lead to efficient computational procedures. For bursty interrupted batch Poisson arrivals, it is shown that an exact numerical analytic analysis is only possible for very small models. For real-life sized systems, an approximation method has been developed.

Multi-queue system with finite shared buffer under Batch Poisson and/or Interrupted Batch Poisson arrivals: a computational analysis

Multi-queue system with finite shared buffer under Batch Poisson and/or Interrupted Batch Poisson arrivals: a computational analysis

A semi-product-form for a pair of queues with finite batches: Equilibrium state probabilities and response time densities

A Markovian network of two queues, with finite size batch Poisson arrivals and departures, is solved approximately, but to arbitrary accuracy, for its equilibrium state probabilities. Below a pair of thresholds on the queue lengths, a modification of the Spectral Expansion Method is used to construct a semi-product-form at all lengths of one queue in a finite lattice strip defined by the threshold of the other queue. No additional special arrival streams are required, for example at empty queues, from which it is already known that a product-form can be constructed. Hence the first exact closed form solution for the equilibrium probabilities in an unmodified Markovian queueing network with batches is obtained, the only constraint being finiteness of the batches. The method is illustrated numerically, first in a tandem network and then in a two-node network with feedback. Simulation results confirm convincingly the precision of the method, and partial batch forwarding and discarding are thereby compared quantitatively. Response time distributions are derived using the generating function method, which complements well the semi-product form of the equilibrium state probabilities. Again, agreement with simulation is excellent. The regenerative simulation method was used, so that no warm-up period was needed, and the statistical estimates for the 95% confidence bands are explained for the different cases of state occupancy and response time prediction at equilibrium.

Computing the Moments of Polling Models with Batch Poisson Arrivals by Transform Inversion

A polling model is defined as a queueing station with a single server serving multiple queues following a certain rule to determine when to stop serving one queue and which queue to serve next. The key to much of the analysis of these systems is to obtain the first two moments of queue lengths at polling instants, which are defined as the instants when the server starts to serve each queue. This study develops a novel approach, in which the generating functions of queue lengths at polling instants are computed iteratively. Transform inversion is then applied to invert these generating functions to obtain their first two moments efficiently. Because the expected waiting time of each queue depends on some basic distributions, such as the service time distributions, the switchover distributions, and the arrival batch distributions only through their first two moments, this study proposes a new transform inversion technique that is called “the pseudotransform inversion algorithm” by introducing “pseudotransforms” of these distributions, which are simply approximations to the transforms with Taylor’s expansions that agree up to a certain order. Then, the expected waiting times can be computed using transform inversion, because they are simply functions of two moments. This study also discusses the special case involving zero switchover times. The strongest benefit of the pseudotransform inversion algorithm over current approaches is that it significantly decreases the computation effort when the number of queues is large. Several numerical examples are devised to validate our approach and show its efficiency in analyzing polling models. The results show that the pseudotransform inversion algorithm is indeed quite efficient for analyzing polling stations. Although polling models are the best example that we have found to date, in any situation where the quantity to be computed depends only on the first several moments of the input distributions, the pseudotransform inversion algorithm holds promise as a computational algorithm.

Burst ratio in the finite-buffer queue with batch Poisson arrivals

We study the burst ratio in the queueing system with finite buffer and batch arrivals. The study is motivated by computer networking, in which packet losses occur due to queueing mechanisms and buffer overflows. First, we derive the formula for the burst ratio in the case of compound Poisson arrivals, general distribution of the service time and general distribution of the batch size. Then, we study its asymptotic behavior, as the buffer size grows to infinity. Using the obtained analytical solutions, we present several numerical examples with various batch size distributions, service time distributions, buffer sizes and system loads. Finally, we compare the computed burst ratios with values obtained in simulations.

Sojourn times in different modes in the queueing system with batch Poisson arrivals, general service times, and hysteretic policy

Consideration is given to the model of the SIP server modelled as a single-server queueing system with finite capacity R and two batch Poisson flows (priority and non-priority), general service times, and bi-level hysteretic control of arrival rates, which is defined by two numbers: L and H (0 < L < H < R). The hysteretic policy under consideration implies that, at each instant, the system can be in one of the three operation modes: normal, overload, and blocking. The system accepts customers from both flows when it is in normal mode, accepts only priority flow when is it overloaded, and rejects all new customers in the discard modes. It is assumed that the switching between operation modes can occur only at the instant of service completion. The analytical method for computing the distributions (in terms of Laplace–Stieltjes transforms) of system’s sojourn times in each operation mode and return times to the normal mode is proposed. Illustative examples are presented.

AnMX/G/1 queueing system with disasters and repairs under a multiple adapted vacation policy

We consider a queueing system with batch Poisson arrivals subject to disasters which occur independently according to a Poisson process but affect the system only when the server is busy, in which case the system is cleared of all customers. Following a disaster that affects the system, the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. The stationary regime of this process is analyzed using the supplementary variables method. We obtain the probability generating function of the number of customers in the system, the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. The stability condition for the system and the Laplace transform of the time between two consecutive disasters affecting the system is obtained by analyzing an embedded Markov renewal process. The statistical characteristics of the batches that complete service without being affected by disasters and those of the partially served batches are also derived. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 171–189, 2015

Analysis of fibre-loop optical buffers with a void-avoiding schedule

In optical packet/burst switching, fibre-loop optical buffers provide a compact and effective means of contention resolution. In case of fixed packet length, the involved loop length is typically chosen matched (equal to the packet length), and the loops are arranged in parallel, constituting a single-stage buffer. In this contribution, we investigate the performance of such a buffer in an asynchronous network setting, assuming batch-Poisson arrivals and assuming a so-called void-avoiding schedule. We show that by time-discretisation, the fibre-loop dynamics can be captured by a particular type of exhaustive polling model. We obtain performance measures such as the moments of the optical queue content and packet delay for the discretised model as well as for the asynchronous optical buffer. We illustrate our approach by various numerical examples.

On transient queue-size distribution in the batch-arrivals system with a single vacation policy

A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.

Transient Analysis of a Single-Server Queue with Batch Arrivals Using Modeling and Functions Akin to the Modified Bessel Functions

Abstract — The paper considers a single-server queue with fixed-size batch Poisson arrivals and exponential service times, a model that is useful for a buffer that accepts messages arriving as fixed size batches of packets and releases them one packet at time. Transient performance measures for queues have long been recognized as being complementary to the steady-state analysis. The focus of the paper is on the use of the functions that arise in the analysis of the transient behaviour of the queuing system. The paper exploits practical modelling to obtain a solution to the integral equation encountered in the analysis. Results obtained indicate that under heavy load conditions, there is significant disparity in the statistics between the transient and steady state values. Keywords — batch arrivals, modelling, single-server queue, time-varying probabilities, transient analysis. I. system. I NTRODUCTION H E paper considers a single-server queue with fixed-size Poisson arrivals and exponential service times. This model is useful for a buffer that accepts messages arriving in fixed size batches of packets and releases one packet at time [1]. Justifications for the transient analysis of traffic in telecommunication systems abound [2]. It has long been recognized that transient performance measures for queues are complementary to the steady-state results [3-6] because there often exists a need to understand the initial behaviour of a system. In general queuing systems, arrivals at a service point (e.g. a switch) may occur in batches of different sizes. Due to congestion these arrivals may be queued for later forwarding to the destination. This paper considers the case in which the arrivals are of fixed size, and occur in continuous time. There may be cases where the network traffic is diverted suddenly to cope with faults, as in automatic protection switching (APS) in which the transport system re-directs traffic when faults and failures occur in subcomponents of the network. In such cases a service point may experience a sudden increase in its load, and this may continue until the original fault has been cleared. After the fault is corrected, traffic reverts to the previous distribution, and this presents another perturbation in the network. There exists now a reduction in the load in parts of the network,

Sojourn time analysis of a queueing system with two‐phase service and server vacations

AbstractWe consider a two‐phase service queueing system with batch Poisson arrivals and server vacations denoted by MX/G1‐G2/1. The first phase service is an exhaustive or a gated bulk service, and the second phase is given individually to the members of a batch. By a reduction to an MX/G/1 vacation system and applying the level‐crossing method to a workload process with two types of vacations, we obtain the Laplace–Stieltjes transform of the sojourn time distribution in the MX/G1‐G2/1 with single or multiple vacations. The decomposition expression is derived for the Laplace–Stieltjes transform of the sojourn time distribution, and the first two moments of the sojourn time are provided. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007

A pseudoconservation law for a time-limited service polling system with structured batch poisson arrivals

We consider a cyclic-service queueing system (polling system) with time-limited service, in which the length of a service period for each queue is controlled by a timer, i.e., the server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then proceeds to the next queue. The customer whose service is interrupted due to the timer expiration is attended according to the nonpreemptive service discipline. For the cyclic-service system with structured batch Poisson arrivals ( M x/G /1) and an exponential timer, we derive a pseudoconservation law and an exact mean waiting time formula for the symmetric system.

Factorization and Stochastic Decomposition Properties in Bulk Queues with Generalized Vacations

This paper considers a class of stationary batch-arrival, bulk-service queues with generalized vacations. The system consists of a single server and a waiting room of infinite capacity. Arrivals of customers follow a batch Markovian arrival process. The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in groups of fixed size B. For this class of queues, we show that the vector probability generating function of the stationary queue length distribution is factored into two terms, one of which is the vector probability generating function of the conditional queue length distribution given that the server is on vacation. The special case of batch Poisson arrivals is carefully examined, and a new stochastic decomposition formula is derived for the stationary queue length distribution.

Characterizing losses during busy periods in finite buffer systems

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ&gt; 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

Characterizing losses during busy periods in finite buffer systems

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ = 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and when ρ &amp;gt; 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, when ρ = 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

Study on optimal queue-length-threshold scheduling policy for an ATM multiplexer with finite buffers and batch Poisson arrivals

This paper studies an optimal queue-length-threshold (QLT) scheduling policy for an ATM multiplexer with two types of traffic: real-time and nonreal-time. The proposed QLT scheduling policy is that if the number of queued nonreal-time packets is greater than or equal to a threshold value α, c2 servers are allocated to the nonreal-time traffic and the rest c1(=c−c2) servers are scheduled with priority policy that the real-time are given higher priority over the nonreal-time; otherwise all the c servers are scheduled with priority policy. The arrival process for each type of traffic is assumed to be batch Poisson with arbitrarily positive integer-valued probability distributions so as to describe the different bandwidth-requirement traffic from narrowband (NB) to wideband (WB). The multiplexer is modeled as a multiple synchronous-server system with separate finite buffers for each type of traffic. Performance measures are exactly derived by using a two-dimensional imbedded Markov-chain approach and the transition probabilities are exactly obtained via the application of residue theorem in complex variable. A trade-off between the effective measures of the two types of traffic is discussed under different (c2, α). Some performance criteria are defined to intend to determine an optimal QLT scheduling policy. This optimal QLT policy can indeed improve the performance criterion above 2–3% over the priority policy.

Performance of a token-passing system with batch arrivals and its application to file transfers

This paper investigates the performance of token-passing systems with limited service and Poisson arrivals. For pure Poisson arrivals, the Laplace-Stieltjes Transform (LST) of an approximate customer/packet waiting time distribution is derived and expressed as a functional equation, from which the approximate mean and variance of waiting time are obtained; for batch Poisson arrivals, an approximate mean of waiting time is derived. Mean waiting time approximations are compared against both simulation results and other results presented in the literature; the variance approximation is evaluated by comparing it to simulation results only, since no other results on variance have been reported so far. As application examples of our analytical results, multimedia file transfers over token-passing LANs are modelled and studied.