Delay and overflow of discrete-time priority queue with burst arrivals and partial buffer sharing

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

This paper presents a traffic model for an ATM multiplexer providing video, voice, image, and data services. The traffic model classifies the input traffic into two types: real-time and non-real-time. The input process for real-time traffic is a Markov modulated Bernoulli process (MMBP), while that for non-real-time traffic is a Bernoulli process with batches. The z-transform of the cell interarrival time A/sub MMBP/(z) is obtained. The average traffic load /spl rho//sub MMBP/ and the squared coefficient of variation of the time between success cell C/sup 2//sub MMBP/ are then obtained from the respective derivatives of A/sub MMBP/(z). The autocorrelation coefficient of the cell interarrival time with lag 1 /spl psi/, can be also obtained. This paper presents an iterative algorithm that can be used to calculate the MMBP probability distribution of the cell interarrival time. So far few results about discrete-time finite buffer queues with correlated inputs are known. This paper analyzes the performance of an ATM multiplexer considering the correlation and burst in cell arrivals. For the MMBP-stream, the waiting time distribution and the cell loss probability for MMBP+M/sup [X]//D/1/K finite buffer system are all exactly calculated with the use of a recursive algorithm. Calculations show the influence of burstiness and correlation on both the loss probability and waiting time. The accuracy of the analytical results has been largely validated by means of a simulation approach.

Simultaneous optimization of flow-control and scheduling in a single server queue with two job classes: Numerical results and approximation

Metamodeling and multi-objective optimization for buffer allocation in multi-product serial production lines

This paper considers subexponential asymptotics of the tail distributions of waiting times in stationary work-conserving single-server queues with multiple Markovian arrival streams, where all arrival streams are modulated by the underlying Markov chain with finite states and service time distributions may differ for different arrival streams. Under the assumption that the equilibrium distribution of the overall (i.e., customer-average) service time distribution is subexponential, a subexponential asymptotic formula is first shown for the virtual waiting time distribution, using a closed formula recently found by the author. Further when customers are served on a FIFO basis, the actual waiting time and sojourn time distributions of customers from respective arrival streams are shown to have the same asymptotics as the virtual waiting time distribution.

We consider a single-server queue with renewal input and Markovian service process where server serves customers in batches according to a general bulk service rule. Queue length distributions at pre-arrival and arbitrary epochs have been obtained along with some important performance measures such as probability of blocking, mean queue lengths and mean waiting times. The analysis has been carried out assuming finite-buffer space for the arriving customers. The model has potential applications in areas such as computer networks, telecommunication systems and manufacturing systems.

We study a single server queue with finite buffer that may represent an ATM multiplexer. The input traffic is a superposition of independent heterogeneous batch Markovian arrival processes. Service durations are supposed to be exponential. We derive the loss probability and the delay and jitter distributions for each individual process, as well as for the superposed one. We show that there is a significant difference between performances of individual and global flows, justifying our work. The approach is based on martingale techniques. >

It is well known that queues with exponentially distributed service times have the smallest Shannon capacity among all single-server queues with the same service rate. In this paper, we study the capacity of timing channels in which the service time distributions have bounded support, i.e., Bounded Service Timing Channels (BSTC). We derive an upper bound and two lower bounds on the capacity of such timing channels. The tightness of these bounds is investigated analytically as well as via simulations. We find that the uniform BSTC serves a role for BSTCs that is similar to what the exponential service timing channel does for the case of timing channels with unbounded service time distributions. That is, when the length of the support interval is small, the uniform BSTC has the smallest capacity among all BSTCs.

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) resultP(a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

We focus on correlated queues where service and inter-arrival times are strongly correlated, and the input and output link speeds are the same. We present an analytical method to derive the LST (Laplace Stieltjes transform) of the (actual) waiting time distribution. To investigate the effect of correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where inter-arrival time and service time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlations have significant impact on the system performance.

A discrete time queueing model for the performance of an ATM system is analyzed using matrix analytic methods. Time is segmented into slots with each slot equal to the transmission time of one ATM cell. The ATM system is modeled as a single server queue with Markovian arrivals and service time equal to one slot. The arrival process includes as a special case the superposition of on-off sources, possibly heterogeneous. The queueing model is of the M/G/1 type. By exploring the structure of the M/G/1 type Markov chain the complexity of the solution to the problem is reduced to only the inversion of a 2/spl times/2 matrix irrespective of the size of the Markov chain. This simplification allows us to solve problems of large sizes. In this paper, we generalize this basic model without losing much of the computational simplifications to the case where there are multiple states corresponding to k (k/spl ges/1) arrivals in a time slot (as opposed to a one-to-one mapping between the number of arrivals and the number of states in the Markov chain) providing a semi-Markov model for the number of arrivals.

Performance analysis of Markovian polling systems with single buffers

A multichannel queueing system with an unlimited queue length is considered. In the system, the service time and time intervals between arrivals of customer batches are characterized by exponential distributions and a hysteretic mechanism is applied to control the input flow intensity. The system receives two independent types of flows of customers one of which is blocked in the overload mode. An algorithm for determination of the stationary distribution of the number of customers and the stationary characteristics (the mean queue length, the mean time of waiting in the queue, the probability of a loss of customers) is proposed. The obtained results are verified with the help of a simulation model developed with the help of the GPSS World tools.