Batch Markovian Arrival Process Research Articles

A finite capacity BMAP K /G K /1 queue with the generalized foreground-background processor-sharing discipline

A queueing system with a batch Markov arrival process, several types of customers, generalized foreground-background processor-sharing discipline with minimal served length, and separate finite buffers for customers of different types or a common finite buffer for customers of all types is studied. Mathematical relations for computing the stationary joint distributions of the number of customers of all types in the system are derived.

BMAP/ G/1/ N queue with vacations and limited service discipline

We consider a finite buffer single server queue with batch Markovian arrival process ( BMAP), where server serves a limited number of customer before going for vacation(s). Single as well as multiple vacation policies are analyzed along with two possible rejection strategies: partial batch rejection and total batch rejection. We obtain queue length distributions at various epochs and some important performance measures. The Laplace–Stieltjes transforms of the actual waiting time of the first customer and an arbitrary customer in an accepted batch have also been obtained.

Optimal multi-threshold control by the BMAP/SM/1 retrial system

A single server retrial system having several operation modes is considered. The modes are distinguished by the transition rate of the batch Markovian arrival process (BMAP), kernel of the semi-Markovian (SM) service process and the intensity of retrials. Stationary state distribution is calculated under the fixed value of the multi-threshold control strategy. Dependence of the cost criterion, which includes holding and operation cost, on the thresholds is derived. Numerical results illustrating the work of the computer procedure for calculation of the optimal values of thresholds are presented.

Threshold control by a single-server retrial queue with batch arrivals and group services

A controlled single-server retrial queueing system is investigated. Customers arrive according to batch Markovian arrival process. The system has several operation modes which are controlled by means of a threshold strategy. The stationary distribution is calculated. Optimization problem is considered and a numerical example is presented.

Recursive filters for partially observable finite Markov chains

In this note, we consider discrete-time finite Markov chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such processes. Recursive equations for these filters are derived by means of simple computations involving conditional expectations. An application to the estimation of parameters of the so-called discrete-time batch Markovian arrival process is outlined.

Recursive filters for partially observable finite Markov chains

In this note, we consider discrete-time finite Markov chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such processes. Recursive equations for these filters are derived by means of simple computations involving conditional expectations. An application to the estimation of parameters of the so-called discrete-time batch Markovian arrival process is outlined.

Delay analysis of downlink IP traffic on UMTS mobile networks

Since wireless networks which can carry high bit rates have become ubiquitous, mobile computing is no longer just spoken about. Mobile computing always implies access through a wireless network to an IP network such as the Internet. In order to understand the performance of such links, we propose an analytic model for the down link delay of IP traffic between the Mobile Gateway Server and the End User in a UMTS mobile network. Traffic arriving at the Gateway Server is considered to be bursty in nature and we use a Batch Markovian Arrival Process (BMAP) to model this arrival process. We model the wireless link itself as a modified multi-state Gilbert–Elliot Markov model which takes into account the number of interfering users and whether the channel experiences Ricean fading or not for what we consider a typical indoor, IP-centric environment. We also account in both the analytical model and the simulation for the Forward Error Correction provided by Turbo coding in UMTS to establish realistic packet retransmission rates. Finally we calibrate and verify the correctness of the model with a discrete event simulator.

An integrated model of packetized VBR teletraffic source for cellular NG All-IP wireless networks

We propose the integrated model of a single packetized variable bit rate (VBR) teletraffic source for cellular next generation (NG) All-IP networks. The proposed model integrates two parts: the mobility model and the teletraffic model. The mobility model captures mobility behavior of a single user and modeled by a Markov chain with finite state space. The teletraffic model captures teletraffic characteristics of call and packet levels and is represented by the superposition of a Markov chain and a discrete-time batch Markovian arrival process (D-BMAP). Mobility and teletraffic models are integrated, such that the whole model is a triply stochastic process for which the D-BMAP framework remains valid. Numerical examples of the proposed model indicates that even in presence of probabilistic dependence between user mobility and its teletraffic demands it is still feasible to estimate the actual value of call and packet level teletraffic demands offered by a single user to cellular NG All-IP networks. Performance evaluation studies show that the proposed model can be used to predict call and packet level performance degradation experienced by applications in cellular NG All-IP networks under different mobility behavior of nomadic users.

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.

Analysis of a BMAP/D/1-Timer Multiplexer

In this paper we introduce and analyze a model of a multiplexer queue with a batch Markovian arrival process and a special, timer based, non-work-conserving service discipline. We show that the embedded process at departures is an M/G/1 type process with proper state partitioning, which can be efficiently analyzed by matrix geometric methods. We derive the expressions to compute the distribution of the waiting time. The paper ends with numerical experiments, and points out some interesting features of the system.

Per Connection Performance Analysis of a Frame-Based TDMA/CDMA MAC Protocol Containing both Reservation and Contention Slots

A mixture of reservation plus contention data slots is now widely implemented in TDMA/ CDMA systems to make a system flexible enough to suit various kinds of packets but preserve the good nature of frame-based protocols. We conducted performance analysis for an arbitrary connection in such a system under the assumption of MMPP (Markov-modulated Poisson process) arrivals. Accessible slot locations of this connection in a frame are made general. Success probability in accessing a contention slot is also made general. We have obtained the system size distribution which can be used to evaluate the performance of various frame-based MAC protocols. The MMPP arrival pattern can be generalized to the BMAP (batch Markovian arrival process) family to further accommodate a broader set of traffic sources.

Tail Probability of Low-Priority Queue Length in a Discrete-Time Priority BMAP/PH/1 Queue

ABSTRACT We investigate the tail probability of the queue length of low-priority class for a discrete-time priority BMAP/PH/1 queue that consists of two priority classes, with BMAP (Batch Markovian Arrival Process) arrivals of high-priority class and MAP (Markovian Arrival Process) arrivals of low-priority class. A sufficient condition under which this tail probability has the asymptotically geometric property is derived. A method is designed to compute the asymptotic decay rate if the asymptotically geometric property holds. For the case when the BMAP for high-priority class is the superposition of a number of MAP's, though the parameter matrices representing the BMAP is huge in dimension, the sufficient condition is numerically easy to verify and the asymptotic decay rate can be computed efficiently.

A Spectral Method for a Nonpreemptive Priority BMAP/G/1 QUEUE

In this paper we consider a nonpreemptive priority queue with two priority classes of customers. Customers arrive according to a batch Markovian arrival process (BMAP). In order to calculate the boundary vectors we propose a spectral method based on zeros of the determinant of a matrix function and the corresponding eigenvectors. It is proved that there are M zeros in a set Ω, where M is the size of the state space of the underlying Markov process. The zeros are calculated by the Durand-Kerner method, and the stationary joint probability of the numbers of customers of classes 1 and 2 at departures is derived by the inversion of the two-dimensional Fourier transform. For a numerical example, the stationary probability is calculated.

Functionals of Markovian Branching D-BMAPS

We consider a stochastic system in which Markovian customer attribute processes are initiated at customer arrivals in a discrete batch Markovian arrival process (D-BMAP). We call the aggregate a Markovian branching D-BMAP. Each customer attribute process is an absorbing discrete time Markov chain whose parameters depend both on the phase transition, of the driving D-BMAP, and the number of arrivals taking place at the customer's arrival instant. We investigate functionals of Markovian branching D-BMAPs that may be interpreted as cumulative rewards collected over time for the various customers that arrive to the system, in the transient and asymptotic cases. This is achieved through the derivation of recurrence relations for expected values and Laplace transforms in the former case, and Little's law in the latter case.

The [formula omitted] tandem queue with losses

Tandem queues of the B M A P / G / 1 / N ¯ → ⋅ / P H / 1 / M type are good models for different fragments of communication systems and networks, so their investigation is interesting for theory and applications. These queues may play an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. Exact analytic analysis of this kind of queues for the cases of infinite and finite input buffers is implemented. Possible correlation and group arrivals are taken into account by means of considering the Batch Markovian Arrival Process ( BMAP) as input stream to the system. The Markov chain embedded at service completion epochs at the first service stage and the process of system states at arbitrary time are investigated. Loss probabilities at the first and second stages are calculated. Numerical results are presented to demonstrate the feasibility of the presented algorithms and describe the performance of the queueing model under study. The necessity of taking the input correlation into account is illustrated.

An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

For a broad class of discrete- and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some multiserver queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the probability generating functions (PGFs) of the stationary numbers in queue and in system, as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with batch Markovian arrival process (BMAP) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Discrete-time modeling of TCP Reno under background traffic interference with extension to RED-based routers

This paper introduces a discrete-time model which captures the essential protocol features of the congestion control mechanism used by the TCP Reno protocol, subject to interference from other sources. Under this model, a single target session is modeled according to the TCP Reno mechanism, including slow start, congestion avoidance, fast retransmit and fast recovery. At the same time, other sources are modeled as a background process using a discrete batch Markov arrival process (D-BMAP). The D-BMAP process has been modified such that the transitions between the phases are dependent on the number of lost packets from the background process. This introduces a feedback process, which can be used to model an aggregation of TCP sources. In order to capture all the TCP Reno protocol features, two levels of Markov process modeling are used: a microscopic level, at the packet transmission time boundaries, and a macroscopic one, at the start of the new transmission windows. In addition, it is shown how the model can be extended to model networks with RED-based routers.Several performance measures are derived, and numerical examples which demonstrate the protocol features are presented.

Analysis of the BMAP/G/1 retrial system with search of customers from the orbit

We consider a single server retrial queuing model in which customers arrive according to a batch Markovian arrival process. Any arriving batch finding the server busy enters into an orbit. Otherwise one customer from the arriving batch enters into service immediately while the rest join the orbit. The customers from the orbit try to reach the service later and the inter-retrial times are exponentially distributed with intensity depending (generally speaking) on the number of customers on the orbit. Additionally, the search mechanism can be switched-on at the service completion epoch with a known probability (probably depending on the number of customers on the orbit). The duration of the search is random and also probably depending on the number of customers in the orbit. The customer, which is found as the result of the search, enters the service immediately if the server is still idle. Assuming that the service times of the primary and repeated customers are generally distributed (with possibly different distributions), we perform the steady state analysis of the queueing model.

Traffic splitting in a network: split traffic models and applications

The contemporary high-speed networks, e.g. the Internet and asynchronous transfer mode (ATM) networks provide a convenient and cost-effective communication platform to carry the emerging multimedia applications. However, problems, such as network congestion caused by overloaded traffic get worse day by day. Thus, how to properly control the system while maintaining the quality of service (QoS) for users becomes an important and challenging issue. To reach the goal of QoS guarantee, load sharing or traffic balancing is one of important techniques to improve the whole network performance. In this paper, we focus on the probabilistic routing policy and analyze split traffic models under such a probabilistic routing policy with input traffic models of the batch Markovian arrival process (BMAP) and discrete-time BMAP (D-BMAP) which can be employed to characterize the multimedia traffic in current and future networks. We successfully show that the split traffic on each routing path from an original BMAP (D-BMAP) input is still a BMAP (D-BMAP). The detailed traffic structure is also acquired. Applying traffic splitting along with the output process characterization, we propose a network-wise performance evaluation method suitable for mesh networks with a probabilistic routing algorithm to get the end-to-end delay, delay variance, and cell loss probability. With numerical experiments, we show that (i) the probabilistic routing policy with a random routing matrix performs well for systems; (ii) the proposed method of network-wise performance analysis has good accuracy.

Performance evaluation of prioritized scheduling with buffer management for differentiated services architectures

Differentiated services (DiffServ) is an architecture for the Internet in which various applications are supported using a simple classification scheme. Packets entering the DiffServ domain are marked depending on the packets' class. In this paper we introduce a versatile service and buffering scheme for different classes in a differentiated services Internet. The objective of this scheme is to introduce relative differentiation in terms of mean delay, throughput. In this scheme both high-buffer-priority (HPC) and low-buffer-priority (LPC) classes have reserved buffers. In addition, they have a shared buffer, where the priority of the shared buffer occupancy is to HPC traffic. Service priorities can be for high-buffer-priority traffic, low-buffer-priority traffic or round robin. An exact performance model for the proposed scheme is introduced. The performance model represents HPC and LPC traffic arrivals by a discrete batch Markov arrival process (D-BMAP). The model is used to obtain loss ratios, packet delays and throughputs for both HPC and LPC traffic. Performance results show that implementing buffer priorities, with a weighted round robin packet scheduling policy, results in low packet loss ratio and/or delay for the high-priority class without starving the low-priority class.