Showing papers in "Advances in Applied Probability in 1992"

On Some Exponential Functionals of Brownian Motion

Abstract: In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T]of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.

Stability of Markovian processes. I : Criteria for discrete-time chains

Abstract: In this paper we connect various topological and probabilistic forms of stability for discrete-time Markov chains. These include tightness on the one hand and Harris recurrence and ergodicity on the other. We show that these concepts of stability are largely equivalent for a major class of chains (chains with continuous components), or if the state space has a sufficiently rich class of appropriate sets ('petite sets'). We use a discrete formulation of Dynkin's formula to establish unified criteria for these stability concepts, through bounding of moments of first entrance times to petite sets. This gives a generalization of Lyapunov-Foster criteria for the various stability conditions to hold. Under these criteria, ergodic theorems are shown to be valid even in the non-irreducible case. These results allow a more general test function approach for determining rates of convergence of the underlying distributions of a Markov chain, and provide strong mixing results and new versions of the central limit theorem and the law of the iterated logarithm.

Limit distributions of maximal segmental score among markov-dependent partial sums

Abstract: Let s1, " , sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process S.,m = 1E' Xss,si+,, m = 1, 2, - - - is determined by a realization {s})=o of states with so = au and the real-valued i.i.d. bounded variables XP associated with the transitions si = a, si+l = fP. Assume Xop has negative stationary mean. The explicit limit distribution of the maximal segmental sum

Random Johnson-Mehl tessellations

Abstract: A unified exposition of random Johnson–Mehl tessellations in d-dimensional Euclidean space is presented. In particular, Johnson-Mehl tessellations generated by time-inhomogeneous Poisson processes and nucleation-exclusion models are studied. The ‘practical' cases d = 2 and d = 3 are discussed in detail. Several new results are established, including first- and second-order moments of various characteristics for both Johnson–Mehl tesselations and sectional Johnson–Mehl tessellations.

Asymptotic Properties of the Least Squares Method for Estimating Transfer Functions and Disturbance Spectra

Abstract: The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞ . It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

Stochastic order for redundancy allocations in series and parallel systems

Abstract: The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

Resource pooling in queueing networks with dynamic routing

Abstract: In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.

Optimal allocation of resources to nodes of parallel and series systems

Abstract: In this paper we consider parallel and series systems, the components of which can be 'improved'. The 'improvement' consists of supplying the components with cold or hot standby spares or by allotting to them fixed budgets for minimal repairs. A fixed total resource of spares or minimal repairs is available. We find the optimal allocation of the resource items in several commonly encountered settings.

Simulating level-crossing probabilities by importance sampling

Abstract: Let X, X2, ... be independent and identically distributed random variables such that EX, 0) > 0. Fix M > 0 and let T = inf {n:X +X2+, ...+ X, > M} (T = +o, if X1 + X2 + " + X, 5 M for every n = 1, 2, - - .). In this paper we consider the estimation of the level-crossing probabilities P(T - oo, we prove asymptotic optimality results for the simulation of the probabilities P(T < oo) and P(T - Mx). The paper ends with an example.

On ergodicity and recurrence properties of a markov chain with an application to an open jackson network

Abstract: This paper gives an overview of recurrence and ergodicity properties of a Markov chain. Two new notions for ergodicity and recurrence are introduced. They are called p-geometric ergodicity and p-geometric recurrence respectively. The first condition generalises geometric as well as strong ergodicity. Our key theorem shows that p-geometric ergodicity is equivalent to weak p-geometric recurrence. The latter condition is verified for the time-discretised two-centre open Jackson network. Hence, the corresponding two-dimensional Markov chain is p-geometrically and geometrically ergodic, but not strongly ergodic. A consequence of A-geometric ergodicity with A of product-form is the convergence of the Laplace-Stieltjes transforms of the marginal distributions. Consequently all moments converge. GEOMETRIC; STRONG AND p-GEOMETRIC ERGODICITY; FOSTER, POPOV AND DOEBLIN

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

Abstract: The M/G/ 1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/ 1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/ 1 queues with shortest residual processing time discipline.

Two parallel processors with coupled inputs

Abstract: We consider the double queue arising from a system consisting of two processors serving three job streams generated by independent Poisson sources. The central job stream of rate v consists of jobs which place resource demands on both processors, which are handled separately by each processor once the request is made. In addition, the first processor receives background work at a rate of A while the second receives similar tasks at a rate r1. Each processor has exponentially distributed service times with rates ao and P respectively. A functional equation is found for P(z, w), the generating function of the joint queue-length distribution, which leads to a relation between P(z, 0) and P(0, w) in the region Izl, Iwl < 1 of a complex algebraic curve associated with the problem. The curve is parametrized by elliptic functions z(?) and w(?) and the relation between P(z(?), 0) and P(0, w(?)) persists on their analytic continuation as elliptic functions in the a-plane. This leads to their eventual determination by an appeal to the theory of elliptic functions. From this determination we obtain asymptotic limit laws for the expectations of the mean number of jobs in each queue conditioned on the other, as the number of jobs in both processors tends to oo. Transitions are observed in the asymptotic behavior of these quantities as one crosses various boundaries in the parameter space. An interpretation of these results via the theory of large deviations is presented.

A new ordering for stochastic majorization: theory and applications

Abstract: In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.

Event and time averages: a review

Abstract: This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case in discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases

Existence of non-trivial quasi-stationary distributions in the birth-death chain

Abstract: We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

Some notes on two types of minimal repair

Abstract: Two types of minimal repair are discussed. After statistical minimal repair, the state of a system is statistically identical to what it was just before the failure. Physical minimal repair restores the failed unit to its exact physical condition before the failure. Several examples are given.

Analyticity of Poisson-driven stochastic systems

Abstract: Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ, E λ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.

Brownian bridge asymptotics for random mappings

Abstract: The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the Aldous-Pitman (1994) result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random p-mappings.

Generalised eigenproblems arising in aggregated markov processes allowing for time interval omission

Abstract: We consider a continuous-time Markov chain in which one cannot observe individual states but only which of two sets of states is occupied at any time Furthermore, we suppose that the resolution of the recording apparatus is such that small sojourns, of duration less than a constant deadtime, cannot be observed We obtain some results concerning the poles of the Laplace transform of the probability density function of apparent occupancy times, which correspond to a problem about generalised eigenvalues and eigenvectors These results provide useful asymptotic approximations to the probability density of occupancy times A numerical example modelling a calcium-activated potassium channel is given Some generalisations to the case of random deadtimes complete the paper

The efficiency and heavy traffic properties of the score function method in sensitivity analysis of queueing models

Abstract: This paper studies computer simulation methods for estimating the sensitivities (gradient, Hessian etc.) of the expected steady-state performance of a queueing model with respect to the vector of parameters of the underlying distribution (an example is the gradient of the expected steady-state waiting time of a customer at a particular node in a queueing network with respect to its service rate). It is shown that such a sensitivity can be represented as the covariance between two processes, the standard output process (say the waiting time process) and what we call the score function process which is based on the score function. Simulation procedures based upon such representations are discussed, and in particular a control variate method is presented. The estimators and the score function process are then studied under heavy traffic conditions. The score function process, when properly normalized, is shown to have a heavy traffic limit involving a certain variant of two-dimensional Brownian motion for which we describe the stationary distribution. From this, heavy traffic (diffusion) approximations for the variance constants in the large sample theory can be computed and are used as a basis for comparing different simulation estimators. Finally, the theory is supported by numerical results.

Sensitivity analysis for stationary and ergodic queues

Abstract: Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

Stochastic comparisons for queueing models via random sums and intervals

Abstract: We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered. These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

The joint distribution of sojourn times in finite Markov processes

Abstract: Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

A note on the multitype measure branching process

Abstract: The existence of a class of multitype measure branching processes is deduced from a single-type model introduced by Li [8], which extends the work of Gorostiza and Lopez-Mimbela [5] and shows that the study of a multitype process can sometimes be reduced to that of a single-type one.

Extension of the bivariate characterization for stochastic orders

Abstract: The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate functions, g 1(x, y) and g 2(x, y), satisfying certain properties. This characterization allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them.

Light traffic approximations in many-server queues

Abstract: This paper complements two previous studies (Daley and Rolski (1984), (1991)) by investigating limit properties of the waiting time in k-server queues with renewal arrival process under light traffic conditions. Formulae for the limits of the probability of waiting and the waiting time moments are derived for the two approaches of dilation and thinning of the arrival process. Asmussen's (1991) approach to light traffic limits applies to the cases considered, of which the Poisson arrival process (i.e. M/G/k) is a special case and for which formulae are given

The distribution of apparent occupancy times in a two-state markov process in which brief events cannot be detected

Abstract: We consider a two-state Markov process in which the resolution of the recording apparatus is such that small sojourns, of duration less than some constant deadtime r, cannot be observed: the so-called time interval omission problem. We express the probability density of apparent occupancy times in terms of an exponential and infinitely many damped oscillations. Using a finite number of these gives an extremely accurate approximation to the true density for all except small values of the time t.

Diffusion approximations for some simple chemical reaction schemes

Abstract: We shall establish functional limit laws for the concentration of the various species in simple chemical reactions. These results allow us to conclude that, under quite general conditions, the concentration has an approximate normal distribution. We provide estimates for the mean and the variance which are valid at all stages of the reaction, in particular, the non-equilibrium phase. We also provide a detailed comparison of our results with the earlier work of Dunstan and Reynolds ([7], [8]).

The interchangeability of tandem queues with heterogeneous customers and dependent service times

Abstract: Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is c,. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the p,'s. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ./M/1 queues is a special case of this result. Other special cases provide interesting new results.

Exact distribution theory for some point process record models

Abstract: Let Y0, Y1, .Y2, be an i.i.d. sequence of random variables with continuous distribution function, and let P be a simple point process on 0? t 5 o00, independent of the Y,'s. We assume that P has a point at t = 0; we associate Y, with the jth point of P, j h 0, and we say that the Yi's occur at the arrival times of P. Yo is considered a 'reference value'. The first YI (j _ 1) to exceed all previous ones is called the first 'record value', and the time of its occurrence is the first 'record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the first n record times, for general P; in some cases the series can be summed. We find the intensity of the record process when P is a general birth process, and when P is a linear birth process with m immigration sources we find the distribution of the number of records in (0, t]. For m = 0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process when m = 1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. For m = 2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times when P is an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.