John A. Morrison

Asymptotic analysis of a random walk on a hypercube with many dimensions

In nearest neighbor random walk on an n-dimensional cube a particle moves to one of its nearest neighbors (or stays fixed) with equal probability. the particle starts at 0. How long does it take to reach its stationary distribution? in fact, this occurs surprisingly rapidly. Previous analysis has shown that the total variation distance to stationarity is large if the number of steps N is 1/4n log n. This paper derives an explicit expression for the variation distance as n → ∞ in the transition region N ˜ 1/4n log n. This permits the first careful evaluation of a cutoff phenomenon observed in a wide variety of Markov chains. the argument involves Fourier analysis to express the probability as a contour integral and saddle point approximation. the asymptotic results are in good agreement with numerical results for n as small as 100.

Virtual private networks: joint resource allocation and routing design

We consider the resource allocation problem in the design of intranets or virtual private networks (VPNs) that is faced by a service provider, which has service level agreements with various customers to carry their multiservice traffic. The design allocates bandwidth on each link to the VPNs such that, when the traffic of a customer is optimally routed over its VPN, a weighted aggregate measure of carried bandwidth over the service providers infrastructure is maximized, subject to constraints that each VPN carries a specified minimum. Multiplexing is across services and routes within each VPN, but not across VPNs. The traffic modelling is at the flow or call level, with random arrivals and holding times of calls and each call requiring (effective) bandwidth, which is characteristic of the calls service type, on all links in its route. The network is modelled as a multirate loss network. Scalability of the design process, i.e., the ability to handle OC3 and higher rates, is an important contribution, and this is achieved by the systematic use of a refined uniform asymptotic approximation. Our software package VPN DESIGNER incorporates these results. We report on numerical results for problems with up to 8 VPNs on a network with 8 nodes, 24 OC3 links, 5 services and up to 640 routes.

The equivalence between processor sharing and service in random order

In this note we explore a useful equivalence relation for the delay distribution in the G/M/1 queue under two different service disciplines: (i) processor sharing (PS); and (ii) random order of service (ROS). We provide a direct probabilistic argument to show that the sojourn time under PS is equal (in distribution) to the waiting time under ROS of a customer arriving to a non-empty system. We thus conclude that the sojourn time distribution for PS is related to the waiting-time distribution for ROS through a simple multiplicative factor, which corresponds to the probability of a non-empty system at an arrival instant. We verify that previously derived expressions for the sojourn time distribution in the M/M/1 PS queue and the waiting-time distribution in the M/M/1 ROS queue are indeed identical, up to a multiplicative constant. The probabilistic nature of the argument enables us to extend the equivalence result to more general models, such as the M/M/1/K queue and ./M/1 nodes in product-form networks.

A Novel Distributed Power Control Algorithm for Classes of Service in Cellular CDMA Networks

The paper proposes a distributed power control algorithm for integrating heterogeneous transmitting sources, which have a broad range of statistical/burstiness characteristics and quality of service requirements, in wideband cellular CDMA networks. Speech sources with silence detection and data are canonical examples of service classes. Each service class is characterized by on-off transmissions with characteristic probabilities, desired minimum carrier-to-interference ratio and minimum probability with which the latter is required to be satisfied. In the power control algorithm given here, the received power for each service class at each cell is adapted locally based on only local measurements of the mean and also, importantly, the variance of the interference. The algorithm is derived from an asymptotic analysis in which the bandwidth as well as the number of mobiles are large. The analysis leads to a Gaussian approximation to the interference at each cell, which depends on the power levels. The algorithm is remarkable for its simplicity in the decoupling between classes. This is due in part to an attractive product-form in the expression for the dominant term in the asymptotic expansion of the desired power for each service class and cell. A condition for geometric convergence of the adapted power to the ideal power is obtained. The condition is remarkably unburdensome and only slightly more demanding than the condition based on the mean values of source activities. The condition also defines the capacity of the cellular CDMA network.

Optimization and design of network routing using refined asymptotic approximations

The problems of route optimization, and the sizing of virtual paths and explicit routes in wide-area multi-service broadband networks are considered. The problems are formulated at the call-level in the framework of multi-rate, circuit-switched, loss networks, with effective bandwidth encapsulating cell and packet-level behavior. Broadband networks are characterized by links with very large capacities in circuits, and are expected to support many services each having a characteristic bandwidth or rate. Various asymptotic results based on Uniform Asymptotic Approximations (UAA) have previously been obtained to reduce the complexity of the numerical calculations. This paper offers refinements (RUAA) to UAA to the loss probabilities for a single link, as well as their sensitivities to the offered traffic. Network loss probabilities are obtained by solving fixed-point equations. Another system of equations determines the implied costs for services and links, which are used to guide the network optimization. Refined asymptotic approximations to the networks loss probabilities as well as to the implied costs are proposed based on the RUAA. The refinements are crucial for the accurate evaluations of the implied costs. The complexity of these calculations remains bounded as the link capacities and traffic intensities become increasingly large, and the complexity for the implied costs is independent of the number of services. A network design tool TALISMAN has been extended to implement the refined approximations. Numerical examples illustrate the accuracy of the RUAA.

Refined asymptotic approximations to loss probabilities and their sensitivities in shared unbuffered resources

We consider an unbuffered resource having capacity C, which is shared by several different services. Calls of each service arrive in a Poisson stream and request a fixed, integral amount of capacity, which may depend on the service. An arriving call is blocked and lost if there is not enough capacity. Otherwise, the capacity of the call is held for the duration of the call, and the holding period is generally distributed. It is assumed that C and the traffic intensities of the services are commensurately large and asymptotic approximations are obtained for the loss probabilities and their sensitivities to the traffic intensity of each service. These sensitivities are important in optimizing the performance of multiservice, multirate loss networks. Numerical results illustrate the accuracy of the asymptotic approximations. These results show that while prior asymptotic approximations to the loss probabilities are quite accurate, the new approximations are very accurate. Moreover, while the prior asymptotic...

Heavy Traffic Analysis of Two Coupled Processors

We consider two parallel queues. When both are non-empty, they behave as two independent M/M/1 queues. If one queue is empty the server in the other works at a different rate. We consider the heavy traffic limit, where the system is close to instability. We derive and analyze the heavy traffic diffusion approximation for this model. In particular, we obtain simple integral representations for the joint steady state density of the (scaled) queue lengths. Asymptotic and numerical properties of the solution are studied.

Asymptotic Expansions of Moments of the Waiting Time in a Shared-Processor of an Interactive System

An interactive computer system’ service is characterized by the random waiting (or response) time perceived by users. This paper presents a novel solution to the problem of efficiently computing th...

Interacting queues in heavy traffic

We consider a system of parallel queues with Poisson arrivals and exponentially distributed service requirements. The various queues are coupled through their service rates, causing a complex dynamic interaction. Specifically, the system consists of one primary queue and several secondary queues whose service rates depend on whether the primary queue is empty or not. Conversely, the service rate of the primary queue depends on which of the secondary queues are empty.An important special case arises when the service rates satisfy a specific relationship so that the various queues form a work-conserving system. This case is, in fact, equivalent to a so-called Generalized Processor Sharing (GPS) system. GPS-based scheduling algorithms have emerged as popular mechanisms for achieving service differentiation while providing statistical multiplexing gains.We consider a heavy-traffic scenario, and assume that each of the secondary queues is underloaded when the primary queue is busy. Using a perturbation procedure, we derive the lowest-order asymptotic approximation to the joint stationary distribution of the queue lengths, in terms of a small positive parameter measuring the closeness of the system to instability. Heuristic derivations of these results are presented.We also pursue two extensions: (i) the more general work-conserving case where the service rate of a secondary queue may depend on its own length, and is a slowly varying function of the length of the primary queue; and (ii) the non-work-conserving case where the service rate of a secondary queue may depend on its own length, but not on the length of the primary queue.

Blocking Probabilities for an Underloaded or Overloaded Link with Trunk Reservation

A single link in a circuit-switched network is considered. The link has C circuits, R of which are reserved for the primary traffic. Offered calls arrive in independent Poisson streams with mean rates