Treelike queueing networks: asymptotic stationarity and heavy traffic (original) (raw)
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A new view of the heavy-traffic limit theorem for infinite-server queues
Advances in Applied Probability, 1991
This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.
Convergence of a queueing system in heavy traffic with general patience-time distributions
Stochastic Processes and their Applications, 2011
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. c
2014
A many-server heavy-traffic functional law of large numbers is established for the (Gt/GI/st +GI) /Mt open queueing network, with a finite number of queues (the superscript m), non-stationary non-Poisson external arrival processes (the Gt), non-exponential service times (the first GI), timevarying staffing levels (the st), and customer abandonment following nonexponential patience times (the +GI). Upon service completion, customers are either routed to one of the queues in the network or out of the system according to time-dependent probabilities (the Mt). The limit provides support for a previously proposed deterministic fluid approximation and extends a previously established limit for the Gt/GI/st +GI single queue model.
Weak convergence limits for sojourn times in cyclic queues under heavy traffic conditions
Journal of Applied Probability, 2008
We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.
Continuity of Large Closed Queueing Networks with
This paper studies a closed queueing network containing a hub (a state dependent queueing system with service depending on the number of units residing here) and k satellite stations, which are GI/M/1 queueing systems. The number of units in the system, N , is assumed to be large. After service completion in the hub, a unit visits a satellite station j, 1 ≤ j ≤ k, with probability p j , and, after the service completion there, returns to the hub. The parameters of service times in the satellite stations and in the hub are proportional to 1 N. One of the satellite stations is assumed to be a bottleneck station, while others are non-bottleneck. The paper establishes the continuity of the queue-length processes in non-bottleneck satellite stations of the network when the service times in the hub are close in certain sense (exactly defined in the paper) to the exponential distribution.
Weak convergence limits for closed cyclic networks of queues with multiple bottleneck nodes
Journal of Applied Probability, 2012
We consider a sequence of cycles of exponential single-server nodes, where the number of nodes is fixed and the number of customers grows unboundedly. We prove a central limit theorem for the cycle time distribution. We investigate the idle time structure of the bottleneck nodes and the joint sojourn time distribution that a test customer observes at the nonbottleneck nodes during a cycle. Furthermore, we study the filling behaviour of the bottleneck nodes, and show that the single bottleneck and multiple bottleneck cases lead to different asymptotic behaviours.
Waiting Time Asymptotics in the Single Server Queue with Service in Random Order
Queueing Systems, 2004
We consider the single server queue with service in random order. For a large class of heavy-tailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index −ν, it is shown that the waiting time distribution is also regularly varying, with index 1 − ν, and the pre-factor is determined explicitly.
The Annals of Applied Probability
We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. Since one needs to keep track of the remaining processing times of all jobs in the system in order to describe the evolution, a natural state descriptor for an SRPT queue is a measure valued process in which the state of the system at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system. In this work we are interested in studying the asymptotic behavior of the suitably scaled measure valued state descriptors for a sequence of SRPT queuing systems. Gromoll, Kruk, and Puha (2011) have studied this problem under diffusive scaling (time is scaled by r 2 and the mass of the measure normalized by r, where r is a scaling parameter approaching infinity). In the setting where the processing time distributions have bounded support, under suitable conditions, they show that the measure valued state descriptors converge in distribution to the process that at any given time is a single atom located at the right edge of the support of the processing time distribution with the size of the atom fluctuating randomly in time. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measure valued state descriptors converge in distribution to the process that is identically zero. In Puha (2015) for the setting where the processing time distributions have unbounded support and light tails, a nonstandard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit. In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. Results of Puha (2015) suggest that the right scaling for the measure valued process is governed by a parameter c r that is given as a certain inverse function related to the tails of the first moment of the processing time distribution. Using this parameter we consider a novel scaling for the measure valued process in which the time is scaled by a factor of r 2 , the mass is scaled by the factor c r /r and the space (representing the remaining processing times) is scaled by the factor 1/c r. We show that the scaled measure valued process converges in distribution (in the space of paths of measures). In a sharp contrast to results for bounded support and light tailed service time distributions, this time there is no state space collapse and the limiting measures are not concentrated on a single atom. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain R + valued random field which is determined from a single Brownian motion. Along the way we establish convergence of
Asymptotic properties of queuing networks
1997
A new approach to the analysis of asymptotic properties of closed queuing networks with both constant service rates and, in certain cases, load-dependent service rates is developed. The method is based on a decomposition of the generating function of the normalising constant into simpler node functions which are easily inverted term by term. An exact closed form is obtained for the normalising constant in some cases and an approximation, based on an integral formula, in others. These results are applied to model a large computer system with terminals, which is also used to illustrate the main properties of the normalising constant and the system throughput function as the population increases. The authors' method is compared with others in terms of both accuracy and efficiency. Finally, it is indicated how multiclass networks can be handled, essentially by reduction to a collection of single class networks.
Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic
Mathematics of Operations Research, 2009
In a recent paper it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we provide an alternative proof of this result that does not require exponential integrability. The conditions that we impose (in addition to natural heavy traffic and stability assumptions) are precisely the i.i.d. and square integrability requirements on the network primitives that are typically made in heavy traffic analysis.