Spitzer's Identity Research Articles

Heavy Traffic Limits for Queues with Many Deterministic Servers

Consider a sequence of stationary GI/D/N queues indexed by N↑∞, with servers' utilization 1−β/$\sqrt{N}$, β>0. For such queues we show that the scaled waiting times $\sqrt{N}$WN converge to the (finite) supremum of a Gaussian random walk with drift −β. This further implies a corresponding limit for the number of customers in the system, an easily computable non-degenerate limiting delay probability in terms of Spitzer's random-walk identities, and $\sqrt{N}$ rate of convergence for the latter limit. Our asymptotic regime is important for rational dimensioning of large-scale service systems, for example telephone- or internet-based, since it achieves, simultaneously, arbitrarily high service-quality and utilization-efficiency.

Variance Reduction for Simulating Transient GI/G/1 Behavior

A variety of methods for reducing the variance on Monte Carlo estimators of the expected waiting time Wn of the nth customer in a GI/G/1 queue are studied. The ideas involve Spitzer's identity, importance sampling, and sums with stratified or controlled randomized length.

Two-sided bounds of ruin probabilities

Abstract A new method for obtaining two-sided bounds of ruin probabilities is proposed. This is based on the analysis of so-called geometric sums which are sums of i.i.d.r.v.'s and the number of summands is a r.v. also having a geometric distribution. The method uses probabilistic arguents only and does not use analytic relations such as Spitzer identity. It is shown that some well known bounds can be proved with the help of the developed method. Numerical examples are considered.

Quadrature Routines for Ladder Variables

Let $T = \inf\{n \geq 1: S_n > 0\}$ and $H = S_T$ be ladder variables for a random walk $\{S_n\}_{n \geq 1}$ with nonnegative drift. Integral formulas for generating functions and moments of $T, H$ and related quantities are developed. These formulas are suitable for numerical quadrature and should be easier to implement than formulas based on Spitzer's identity when the distribution of $S_n$ is complicated. The approach used makes key use of the Hilbert transform and the main regularity assumption is that some power of the characteristic function for steps of the random walk is integrable.

The spitzer identity for some bivariate processes in queueing theory: an algebraic approach

This note deals with a purely algebraic development of the Spitzer identity for a bivariate Markov process , which is important for random walks, dams and queues. This identity, comprising the results of Kingman on the process , has been known for a long time in an analytic form as a solution of a certain integral equation. The algebraic approach leads to explicit results of interesting structure. In particular, the bivariate distribution of (Yn, Zn) can be expressed in terms of the marginal distributions of the stochastically dependent variables Yn and Zn.

The spitzer identity for some bivariate processes in queueing theory: an algebraic approach

This note deals with a purely algebraic development of the Spitzer identity for a bivariate Markov process, which is important for random walks, dams and queues. This identity, comprising the results of Kingman on the process, has been known for a long time in an analytic form as a solution of a certain integral equation. The algebraic approach leads to explicit results of interesting structure. In particular, the bivariate distribution of (Yn, Zn) can be expressed in terms of the marginal distributions of the stochastically dependent variablesYnandZn.

The spitzer identity for some bivariate processes in queueing theory: an algebraic approach

This note deals with a purely algebraic development of the Spitzer identity for a bivariate Markov process, which is important for random walks, dams and queues. This identity, comprising the results of Kingman on the process, has been known for a long time in an analytic form as a solution of a certain integral equation. The algebraic approach leads to explicit results of interesting structure. In particular, the bivariate distribution of (Yn, Zn) can be expressed in terms of the marginal distributions of the stochastically dependent variablesYnandZn.

A Simple Analytic Proof of the Pollaczek-Wendel Identity for Ordered Partial Sums

In this note we prove an identity due to Pollaczek (1952) and Wendel (1960). The identity describes the distributions of ordered partial sums of independent identically distributed random variables and thus generalizes Spitzer's identity. Our proof follows from a simple analytic argument applying a kind of Wiener-Hopf decomposition. We also give an extension of the Pollaczek-Wendel identity.

The virtual waiting time of the GI/G/1 queue in heavy traffic

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

The virtual waiting time of the GI/G/1 queue in heavy traffic

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {W n i } of a sequence {Q i } of stable GI/G/1 queues, where W n i is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρ i ↑ 1 as i → ∞. Here &rH i is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {W n i } and obtained limit theorems in the three cases when n 1/2(ρ i -1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {W n i } and obtained limits in the two cases when n 1/2(ρ i − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

Wiener-Hopf-Faktorisierung f�r substochastische �bergangsfunktionen in angeordneten R�umen

If (E, \(\mathfrak{B}\) is a measurable space with a reflexive and transitive ordering such that for every pair x, y e E, x≦y or y≦x holds, then it is intuitive to define increasing, decreasing and equal-height transition functions. P(x, dy) is increasing for instance if $$P(x,\{ y:y \leqq x\} ) = 0{\text{ for all }}x\varepsilon E.$$ If P is an arbitrary substochatsic transition function, then it is shown, that $$(I - sP) = \left( {I - \sum\limits_1^\infty {s^k P_k^ - } } \right) \circ \left( {I - \sum\limits_1^\infty {s^k P_k^ \circ } } \right) \circ \left( {I - \sum\limits_1^\infty {s^k P_k^ + } } \right)$$ holds for 0<=s<1 if the Pk*are certain substochastic transition functions. Here the notation P+ indicates that these transition functions are increasing; similarly P0 and P− is to be interpreted. The Pk*are unique. Their probability interpretation is given. The relation to Wiener-Hopf-Factorization and to Spitzer's Identity is explained. The result is applied to the problem of first exit from a set in a Markov Chain and to the factorization of an infinite matrix into an upper and a lower triangular matrix.

The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

Spitzer's Identity and its use in Probability Theory

Spitzer's Identity and its use in Probability Theory