Leslie D. Servi

M/M/1 queues with working vacations (M/M/1/WV)

The classical single server vacation model is generalized to consider a server which works at a different rate rather than completely stops during the vacation period. Simple explicit formulae for the mean, variance, and distribution of the number and time in the system are presented. The distributional results generalize the classical vacation decomposition with no service during a vacation. This model approximates a multi-queue system whose service rate is one of the two speeds for which the fast speed mode cyclically moves from queue to queue with an exhaustive schedule. This work is motivated and illustrated by the analysis of a WDM optical access network using multiple wavelengths which can be reconfigured.

Algorithmic Solutions to Two-Dimensional Birth---Death Processes with Application to Capacity Planning

Capacity planning of modern telecommunication systems using the Erlang-B or Erlang-C models is hampered by the inability of these models to capture critical system characteristics. Two-dimensional birth–death models offer the opportunity to remedy this. The steady state behavior of two-dimensional birth–death processes is found by numerically solving a linear matrix equation whose special structure is exploited to substantially speed its solution. Two detailed applications drawn from telecommunications capacity planning are presented.

Software-based erasure codes for scalable distributed storage

This paper presents a new class of erasure codes, Lincoln Erasure codes (LEC), applicable to large-scale distributed storage that includes thousands of disks attached to multiple networks. A high-performance software implementation that demonstrates the capability to meet these anticipated requirements is described. A framework for evaluation of candidate codes was developed to support in-depth analysis. When compared with erasure codes based on the work of Reed-Solomon and Luby (2000), tests indicate LEC has a higher throughput for encoding and decoding and lower probability of failure across a range of test conditions. Strategies are described for integration with storage-related hardware and software.

Stochastic bounds for queueing systems with limited service schedules

Abstract Motivated by the study of cyclic service queue processor schedules and token ring local area networks, upper and lower stochastic bounds for a GI/G/1 vacation model with limited service are developed. The limited service vacation model is compared with the Bernoulli schedule vacation model. For the case of Poisson arrivals and infinitely divisible vacation durations simple, closed-form expressions are given for upper and lower bounds of the first two moments of the waiting time. Some upper and lower bounds are also derived for cyclic queues with limited service. The quality of the bounds is illustrated through numerical examples.

The M/G/1/K blocking formula and its generalizations to state-dependent vacation systems and priority systems

The formula for the blocking probability for the finite capacity M/G/1/K in terms of the steady state occupancy probability distribution of M/G/1 and the system utilization is known [Keilson, J. Royal Statistical Soc. Serie B, 28 (1966) 190–201]. The validity of this relationship is demonstrated for a broad class of state dependent M/G/1 vacation systems and priority systems. New methods are employed which may also be of interest in their own right.

A distributed Poisson approximation for preempt-resume clocked schedules

Many telecommunication systems with time-critical requirements use a preempt-resume clocked schedule. An approximation to the ergodic distribution of the time to completion of a low-priority task is obtained by treating the priority service time distribution as the limit of compound Poisson distributions. Explicit formulas for the mean and variance that are highly accurate are given. For random clocked loads, a stochastic bound is provided for the discrepancy between the exact and approximate distributions. For deterministic clocked loads, sample path bounds are found. Simulation results are given to demonstrate the accuracy of the model. >

Loss Probabilities of Hand-in Traffic under Various Protocols, I: Models and Algebraic Results

Mobile telephone traffic business demands that a given base station be able to handle both newly originating traffic and ‘hand-in’ traffic from neighbouring base stations, with priority to be given to the latter, existing traffic. Starting from a baseline standard model in which all channels are available to all traffic, this note examines three other models, each with its own access control protocol that allocates a reserve capacity R amongst the Ntot channels to give priority to the hand-in traffic. These different protocols give different loss probabilities for the two types of traffic. All the results are proved on the basis of assuming a Poisson arrival process and independent exponential service times. This paper concentrates on presenting explicit expressions for many of the loss probabilities, some of them given by the Erlang B loss formula or variants, as well as simple approximations and bounds. A companion paper presents results comparing the various models.

Estimating Waiting Times from Transactional Data

Given transactional data consisting of service starting and finishing times in a queuing system with a Poisson arrival process and service in order of arrival, the distributions and moments of waiting times W1, …, WN, of customers within a busy period of length N are computed efficiently by exploiting an embedded Markov chain. All first moments E(Wk ∣ data from busy period of length N) are computed in O(N3) multiplications; finding the distributions of all Wk on a set of n prespecified times requires O(nN3) multiplications; finding all rth order moments needs O(rN4) multiplications. The algorithms described are all new, as fast or faster than all known algorithms, and address a broader range of performance measures than the extant literature.

The preempt-resume M/G/1/N queue with vacation time and limited service discipline

Abstract An M/G/1 queueing system with two priority classes of traffic and Bernoulli feedback for the low priority class is analyzed. For the low priority customers, the distributions of the ergodic time to completion, the first pass time, and the number in the system are first found in the transform domain for the M/G/1 system and both the mean and the variance are displayed. The distributed Poisson approximation is then used to calculate the corresponding quantities when the high priority customers arrive at clock ticks.