Limit theorems for multiphase queueing systems

One gives a limit theorem for the joint distribution of the stationary waiting times of customers in the queues of a multiphase queueing system, functioning in a heavy traffic regime. One proves that the joint distribution function of the waiting times is a solution of a problem with a directional derivative for an elliptic differential equation in a polyhedral angle.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 212–229, 1986.     

Multi-server queueing systems with cooperation of the servers

In this paper, we consider an N-server queueing model with homogeneous servers in which customers arrive according to a stationary Poisson arrival process. The service times are exponentially distributed. Two new customer’s service disciplines assuming simultaneous service of arriving customer by all currently idle servers are discussed. The steady state analysis of the queue length and sojourn time distribution is performed by means of the matrix analytic methods. Numerical examples, which illustrate advantage of introduced disciplines comparing to the classical one, are presented.     

Convexity of single stage queueing systems with bulk arrival

In this paper we study the convexity of the waiting time, workload and the number of jobs in single stage queueing systems with respect to the bulk size of the arrival process. In particular we show that the number of jobs in a single server queueing systemG ^[x ]/GI/1 and in a multiple server queueing systemG ^[x]/M/c with bulk sizesx=(x ₁ ,x ₂ ,x ₃ ,...) is componentwise convex inx. This is in the sense of the sample path convexity introduced in Shaked and Shanthikumar [11]. These results have applications in the stochastic comparison of bulk arrival queueing systems.Research supported in part by NSF grant DDM-9113008.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

Limit theorems for partially hyperbolic systems

We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms. If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory.

     

A queueing system where customers require a random number of servers simultaneously

A closed single node queueing system with multiple classes is analyzed numerically. The node consists of M identical servers fed by a single queue. Each customer of class r, 1 ⩽ r ⩽ M, acquires r servers simultaneously at the beginning of its service. All r servers are released at the same time upon completion of its service. The service time of a class r customer is exponentially distributed with a mean depending on r. This queueing model is analyzed with a view to obtaining performance measures such as throughput, distribution of busy servers, and queue-length distribution.     

Analysis of multi-level queueing systems with servers breakdown by using recursive solution technique

In this paper we construct a multi-level queueing model that alternates between three modes of an operation system. The service times have followed an Erlang type K   distribution with parameter μ$\mu$. Customers arrive in batches according to a time-homogeneous compound Poisson process with mean rate λ$\lambda$ for the batches. Our aim is to give a recursive scheme for the solution of the steady state equations. Next we derive some important measures of performance which may affect the efficiency of the system under consideration such as the expected waiting time per customer, the expected number of customers who arrive to a full system. The expected number of customers will also be calculated. Finally, we can also calculate the efficiency measures of the system by using the recursive results through an example.     

Limit theorems and Markov approximations for chaotic dynamical systems

Summary We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.     

Limit theorems for monotonic particle systems and sequential deposition

We prove spatial laws of large numbers and central limit theorems for the ultimate number of adsorbed particles in a large class of multidimensional random and cooperative sequential adsorption schemes on the lattice, and also for the Johnson–Mehl model of birth, linear growth and spatial exclusion in the continuum. The lattice result is also applicable to certain telecommunications networks. The proofs are based on a general law of large numbers and central limit theorem for sums of random variables determined by the restriction of a white noise process to large spatial regions.     

Limit theorems for Markov random walks with a fixed number of certain transitions

The limit behavior of Markov chains with discrete time and a finite number of states (MCDT) depending on the number n of its steps has been almost completely investigated [1–4]. In [5], MCDT with forbidden transitions were investigated, and in [6], the sum of a random number of functionals of random variables related by a homogeneous Markov chain (HMC) was considered. In the present paper, we continue the investigation of the limit behavior of the MCDT with random stopping time which is determined by a Markov walk plan II with a fixed number of certain transitions [7, 8]. Here we apply a method similar to that of [6], which allows us to obtain, together with some generalizations of the results of [6], a number of new assertions. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 119–130, Perm, 1990.     

Heavy traffic limit theorems for a sequence of shortest queueing systems

A sequence of shortest queueing systems is considered in this paper. We give weak convergence theorems for the queue length and waiting time processes when the sequence of traffic intensities associated with the sequence of shortest queueing systems approaches the critical value (=1) at appropriate rates.Research supported by the National Natural Science Foundation of China.     

Sample-path analysis of general arrival queueing systems with constant amount of work for all customers

We consider a discrete-time queueing system where the arrival process is general and each arriving customer brings in a constant amount of work which is processed at a deterministic rate. We carry out a sample-path analysis to derive an exact relation between the set of system size values and the set of waiting time values over a busy period of a given sample path. This sample-path relation is then applied to a discrete-time $G/D/c$ queue with constant service times of one slot, yielding a sample-path version of the steady-state distributional relation between system size and waiting time as derived earlier in the literature. The sample-path analysis of the discrete-time system is further extended to the continuous-time counterpart, resulting in a similar sample-path relation in continuous time.     

A transient queueing model for Business Office with standby servers

Customers call Business Offices of a telephone company for services and billing information. We have considered a Business Office in which customers are usually serviced by scheduled servers. These scheduled servers are backed up by some standby servers who will answer a call only when the number of calls waiting to be answered is big. Impatient customers may renege. In this paper, we present a transient solution to a queueing model that can be used to help a Business Office manager efficiently determine the optimal numbers of scheduled and standby servers for achieving the designated service objective cost effectively. We have estimated that our model would save each of the 108 Business Office managers 20 minutes per day. Our tests of the model, using real data from randomly selected days, reveal that the model is about 93% accurate.     

Dynamic server allocation for unstable queueing networks with flexible servers

This paper is concerned with the dynamic assignment of servers to tasks in queueing networks where demand may exceed the capacity for service. The objective is to maximize the system throughput. We use fluid limit analysis to show that several quantities of interest, namely the maximum possible throughput, the maximum throughput for a given arrival rate, the minimum arrival rate that will yield a desired feasible throughput, and the optimal allocations of servers to classes for a given arrival rate and desired throughput, can be computed by solving linear programming problems. We develop generalized round-robin policies for assigning servers to classes for a given arrival rate and desired throughput, and show that our policies achieve the desired throughput as long as this throughput is feasible for the arrival rate. We conclude with numerical examples that illustrate the points discussed and provide insights into the system behavior when the arrival rate deviates from the one the system is designed for.