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The pair (W(t), L(t)t
( \frac1Öt W(t), \frac1Öt L(t) )\left( {\frac{1}{{\sqrt t }}W(t), \frac{1}{{\sqrt t }}L(t)} \right)
) conditioned on the event {T>t} is given ast, whereT is the length of the first busy period. A similar result is also given in the situation whent runs over the arrival moments of customers.

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... The asymptotic formula for the virtual waiting time in a GI/M/1 queueing system in heavy traffic is presented in [33]. In [34,35], the behavior of a scaled pair of a virtual waiting time and queue length processes in heavy traffic is investigated. The author of [36] presented numerous transform formulae for a virtual waiting time in a tandem queue with identical service times at both counters. ...

This paper presents heavy traffic limit theorems for the extreme virtual waiting time of a customer in an open queueing network. In this paper, functional limit theorems are proved for extreme values of important probability characteristics of the open queueing network investigated as the maximum and minimum of the total virtual waiting time of a customer, and the maximum and minimum of the virtual waiting time of a customer. Also, the paper presents the previous related works for extreme values in queues and the virtual waiting time in heavy traffic.

The modern queueing theory is one of the powerful tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems, arising in the network theory and communications theory (called open queueing network). We have proved here the theorem on the law of the iterated logarithm (LIL) for the virtual waiting time of a job in an open queueing network under heavy and light traffic conditions. Also, the work presents a survey of papers for the virtual waiting time of a job in heavy traffic. Finally, we present an application of the proved theorems to the technical model from computer network practice.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk
and lk
denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that $\left(\frac{1}{\sigma \sqrt{k}}l_{k},\frac{1}{\sigma \sqrt{k}}w_{k},\frac{1}{\sigma \root{4}\of{k}}(l_{k}-\mu w_{k})|\tau >k\right)\overset \scr{D}\to{\rightarrow}(\mu \xi,\xi ,a\sqrt{\xi}\eta)$ as k→ ∞ , where a is some known constant, ξ =W+(1), η =W(1), W+ and W are independent, W+ is a Brownian meander and W is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

The queue size process (t)0≤t≤t0 of the batch arrival queue MX/M/1 is studied under the condition that the duration of its busy period is larger than t0. Explicit formulas for the transition probabilities are given and the limiting Markov process for t0 → ∞ is investigated. Several properties of this process are considered. Its transition probabilities and moments and the distribution of its minimum are derived and a functional limit theorem for the rescaled process is proved. © 1994 John Wiley & Sons, Inc.

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer
networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic
in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (
w, v, u
) is asymptotically stationary in some sense then (
l
,
w, v, u
) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M / G /1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.
Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M / G /1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.
The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

The paper deals with the asymptotic behaviour of queues for which the generic sequence is not necessarily stationary but is asymptotically stationary in some sense. The latter property is defined by an appropriate type of convergence of probability distributions of the sequences to the distribution of a stationary sequence We consider six types of convergence of to
The main result is as follows: if the sequence of the distributions converges in one of six ways then the sequence of distributions of the sequences converges in the same way, independently of initial conditions. Furthermore the limiting distribution is the same as the limiting distribution obtained by the weak convergence of the distributions Here w k and w ∗ k denote the waiting time of the k th unit in the queue generated by ( v, u ) and ( v ⁰ , u ⁰ ) respectively.

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (w,v,u) is asymptotically stationary in some sense then (l,w,v,u) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M / G /1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.
Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M / G /1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.
The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

We prove that in the queueing system GI/G/1 with traffic intensity one, the virtual waiting time process suitably scaled, normed and conditioned by the event that the length of the first busy period exceeds n converges to the Brownian meander process, as n ?8.

Let $\{X_k: k \geqq 1\}$ be a sequence of i.i.d.rv with $E(X_i) = 0$ and $E(X_i^2) = \sigma^2, 0 < \sigma^2 < \infty$. Set $S_n = X_1 + \cdots + X_n$. Let $Y_n(t)$ be $S_k/\sigma n^\frac{1}{2}$ for $t = k/n$ and suitably interpolated elsewhere. This paper gives a generalization of a theorem of Iglehart which states weak convergence of $Y_n(t)$, conditioned to stay positive, to a suitable limiting process.