On the M_t/M_t/K_t+M_t queue in heavy traffic

The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained for the periodic number-of-customers processes under the fluid and diffusion scalings. Other results concern limits for general time-dependent queues and for time-homogeneous queues in steady state.     

Compound Transient Phenomena in Multiphase Queuing Systems. III

The author continues the work on functional limit theorems in multiphase queuing systems (QS) under heavy traffic. In this paper there are proved theorems for the waiting time of a job when at phases of a system various conditions of heavy traffic are satisfied (compound transient phenomena).     

Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse

Certain diffusion processes known as semimartingale reflecting Brownian motions (SRBMs) have been shown to approximate many single class and some multiclass open queueing networks under conditions of heavy traffic. While it is known that not all multiclass networks with feedback can be approximated in heavy traffic by SRBMs, one of the outstanding challenges in contemporary research on queueing networks is to identify broad categories of networks that can be so approximated and to prove a heavy traffic limit theorem justifying the approximation. In this paper, general sufficient conditions are given under which a heavy traffic limit theorem holds for open multiclass queueing networks with head-of-the-line (HL) service disciplines, which, in particular, require that service within each class is on a first-in-first-out (FIFO) basis. The two main conditions that need to be verified are that (a) the reflection matrix for the SRBM is well defined and completely- S, and (b) a form of state space collapse holds. A result of Dai and Harrison shows that condition (a) holds for FIFO networks of Kelly type and their proof is extended here to cover networks with the HLPPS (head-of-the-line proportional processor sharing) service discipline. In a companion work, Bramson shows that a multiplicative form of state space collapse holds for these two families of networks. These results, when combined with the main theorem of this paper, yield new heavy traffic limit theorems for FIFO networks of Kelly type and networks with the HLPPS service discipline. This revised version was published online in June 2006 with corrections to the Cover Date.     

On extreme values in open queueing networks

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.     

About the Sojourn Time Process in Multiphase Queueing Systems

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, asembly lines, etc.). In this paper, we deal with approximations of MQS and present a heavy traffic limit theorems for the sojourn time of a customer in MQS. Functional limit theorems are proved for the customer sojourn time – an important probability characteristic of the queueing system under conditions of heavy traffic.      

A network of priority queues in heavy traffic: One bottleneck station

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.     

Some analytical results for congestion in subscriber line modules

In modern telephone exchanges, subscriber lines are usually connected to the so-called subscriber line modules. These modules serve both incoming and outgoing traffic. An important difference between these two types of calls lies in the fact that in the case of blocking due to all channels busy in the module, outgoing calls can be queued whereas incoming calls get busy signal and must be re-initiated in order to establish the required connection. The corresponding queueing model was discussed recently by Lederman, but only the model with losses has been studied analytically. In the present contribution, we study the model which takes into account subscriber retrials and investigate some of its properties such as existence of stationary regime, derive explicit formulas for the system characteristics, limit theorems for systems under high repetition intensity of blocked calls and limit theorems for systems under heavy traffic.     

Asymptotic relations in queueing theory

For the GI?G?1 queueing system a number of asymptotic results are reviewed. Discussed are asymptotics related to the time parameter for t → ∞ relaxation times, heavy traffic theory, restricted accessibility with large bounds, approximation by diffusion processes, exponential and regular variation of the tail of the waiting time distribution, limit theorems and extreme value theorems.     

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.     

Compound transient phenomena in multiphase queuing systems. II

The paper continues the article by the author on functional limit theorems in queuing systems under heavy traffic. Theorems are proved for the virtual process of serving jobs when at phases of the queuing system various conditions of heavy traffic are satisfied (compound transient phenomena). Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 343–356, July–September, 1999. Translated by Z. Kryžius     

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little??s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.     

On the Full Idle Time in Multiphase Queueing Systems

The modern queueing theory is a powerful tool 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 (such as the so-called multiphase queueing systems, tandem queues, or series of queueing systems). We present heavy traffic limit theorems for the full idle time in multiphase queueing systems. We prove functional limit theorems for values of the full idle time of a queueing system, which is its important probability characteristic. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 367–386, July–September, 2005.     

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.     

多类顾客多服务台队列网络的高负荷极限定理

多类顾客多服务台队列网络广泛地应用到计算机网络、通讯网络和交通网络 .由于系统的复杂性 ,其数量指标的精确解很难求出 .为了寻求逼近解 ,本文用概率测度弱收敛理论对进行了研究 ,在高负荷的条件下 ,我们获得了网输入过程、闲时过程和负荷过程的极限定理 .     

Incompressible limit of solutions of multidimensional steady compressible Euler equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.     

On the fluid limit of the M/G/∞ queue

The paper deals with the fluid limits of some generalized M/G/∞ queues under heavy-traffic scaling. The target application is the modeling of Internet traffic at the flow level. Our paper first gives a simplified approach in the case of Poisson arrivals. Expressing the state process as a functional of some Poisson point process, an elementary proof for limit theorems based on martingales techniques and weak convergence results is given. The paper illustrates in the special Poisson arrivals case the classical heavy-traffic limit theorems for the G/G/∞ queue developed earlier by Borovkov and by Iglehart, and later by Krichagina and Puhalskii. In addition, asymptotics for the covariance of the limit Gaussian processes are obtained for some classes of service time distributions, which are useful to derive in practice the key parameters of these distributions.     

Convergence of a queueing system in heavy traffic with general patience-time distributions

We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.     

The multitype branching diffusion

The multitype branching diffusion (MBD) is considered. A review of the general theory of multitype point processes is given in Section 2, and spatial central limit theorems for homogeneous infinitely divisible processes are proven in Section 3. In Section 4, the MBD is defined, and equations for its first four factorial moment density functions are found. The behaviour of the mean and covariance functionals as time approaches infinity is studied. The MBD with immigration (MBDI) is introduced in Section 5. The existence of a steady state is proven, and spatial central limit theorems are developed for the MBDI.     

Busy period distributions in the steady state single-server queue and their applications

A method of generating probability distributions of the number of customers served in a busy period of a steady state single-server queueing system with univariate and multivariate inputs is described. A few moments in terms of the moments of the input distributions are derived. Some applications of the busy period distributions to such areas as branching processes, traffic flows, first passage problems, ballot theorems and waiting time distributions are briefly mentioned.