Exponential Bounds for Queues with Markovian Arrivals

Exponential bounds P[queue ≥ b] ≤ φe^(-γb) are found for queues whose increments are described by Markovian Additive Processes. This is done application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant γ is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor φ

Exponential Upper Bounds via Martingales for Multiplexers with Markovian Arrivals.

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {O, 1}, with integral service rate. The bound is of the form P[queue length ≥ b] ≤ cy^(-b) for any b ≥ 1 where c 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations

Rigorous Bounds for Loss Probabilities in Multiplexers of Discrete Heterogenous Markovian Sources

Exponential upper bounds of the form P[queue ≥ b] ≤ φy^(-b) are obtained for the distribution of the queue length in a model of a multiplexer in which the input is a heterogeneous superposition of discrete Markovian on-off sources. These bounds are valid at all queue lengths, rather than just asymptotic in the limit b→∞. The decay constant y is found by numerical solution of a single transcendental equation which determines the effective bandwidths of the sources in the limit b→∞. The prefactor φ is given explicitly in terms of y. The bound provides a means to determine rigorous corrections to effective bandwidths for multiplexers with finite buffers

Estimation of buffer overflow probabilities and economies of scale in ATM multiplexers by analysis of a model of packetized voice traffic

We obtain upper bounds on the probability of buffer overflow for an ATM multiplexer of L identical packetized voice sources. The multiplexer is modelled by a FCFS single server queue. The arrivals at the multiplexer are a homogenous superposition of the arrivals from L independent identical sources, with each source modelled by a copy of a discrete time Markov Chain which we call the Cell Level Model. Throughout, appropriate parameters are scaled with L, to maintain a constant load over all superposition sizes. The probability that, the queue-length (q^) of the queue in a finite buffer exceeds the buffer size b, is bounded above by the probability that the queue-length (qL) of the queue m an infinite buffer exceeds length b In order to bound the former above, we find upper bounds or approximations for the latter by using the theory of, • Large Deviations, to determine its asymptotics for large b, • Martingales, to obtain upper bounds, valid for all positive b, • Large Deviations, to determine its asymptotics for large L for time rescaled (proportional to L) arrival processes. These demonstrate the multiplexing gam and economies of scale obtainable from large and small buffers and large multiplexers, respectively

Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe

A tandem queue with Lévy input: a new representation of the downstream queue length.

In this paper we present a new representation for the steady state distribution of the workload of the second queue in a two-node tandem network. It involves the difference of two suprema over two adjacent intervals. In case of spectrally-positive

Two extensions of Kingman's GI/G/1 bound

A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass $\Sigma\textrm{GI/G/1}$ queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes

Investigation of delay jitter of heterogeneous traffic in broadband networks

Scope and Methodology of Study: A critical challenge for both wired and wireless networking vendors and carrier companies is to be able to accurately estimate the quality of service (QoS) that will be provided based on the network architecture, router/switch topology, and protocol applied. As a result, this thesis focuses on the theoretical analysis of QoS parameters in term of inter-arrival jitter in differentiated services networks by deploying analytic/mathematical modeling technique and queueing theory, where the analytic model is expressed in terms of a set of equations that can be solved to yield the desired delay jitter parameter. In wireless networks with homogeneous traffic, the effects on the delay jitter in reference to the priority control scheme of the ARQ traffic for the two cases of: 1) the ARQ traffic has a priority over the original transmission traffic; and 2) the ARQ traffic has no priority over the original transmission traffic are evaluated. In wired broadband networks with heterogeneous traffic, the jitter analysis is conducted and the algorithm to control its effect is also developed.Findings and Conclusions: First, the results show that high priority packets always maintain the minimum inter-arrival jitter, which will not be affected even in heavy load situation. Second, the Gaussian traffic modeling is applied using the MVA approach to conduct the queue length analysis, and then the jitter analysis in heterogeneous broadband networks is investigated. While for wireless networks with homogeneous traffic, binomial distribution is used to conduct the queue length analysis, which is sufficient and relatively easy compared to heterogeneous traffic. Third, develop a service discipline called the tagged stream adaptive distortion-reducing peak output-rate enforcing to control and avoid the delay jitter increases without bound in heterogeneous broadband networks. Finally, through the analysis provided, the differential services, was proved not only viable, but also effective to control delay jitter. The analytic models that serve as guidelines to assist network system designers in controlling the QoS requested by customer in term of delay jitter