A multiclass feedback queue in heavy traffic

1988 ◽  
Vol 20 (01) ◽  
pp. 179-207 ◽  
Author(s):  
Martin I. Reiman
Keyword(s):  
Random Number ◽  
Queueing System ◽  
Heavy Traffic ◽  
Arrival Process ◽  
Service Discipline ◽  
Reflected Brownian Motion ◽  
One Dimensional ◽  
Heavy Traffic Limit ◽  
Single Station ◽  
Feedback Queue

We consider a single station queueing system with several customer classes. Each customer class has its own arrival process. The total service requirement of each customer is divided into a (possibly) random number of service quanta, where the distribution of each quantum may depend on the customer's class and the other quanta of that customer. The service discipline is round-robin, with random quanta. We prove a heavy traffic limit theorem for the above system which states that as the traffic intensity approaches unity, properly normalized sequences of queue length and sojourn time processes converge weakly to one-dimensional reflected Brownian motion.

A multiclass feedback queue in heavy traffic

1988 ◽  
Vol 20 (1) ◽  
pp. 179-207 ◽  
Author(s):  
Martin I. Reiman
Keyword(s):  
Random Number ◽  
Queueing System ◽  
Heavy Traffic ◽  
Arrival Process ◽  
Service Discipline ◽  
Reflected Brownian Motion ◽  
One Dimensional ◽  
Heavy Traffic Limit ◽  
Single Station ◽  
Feedback Queue

We consider a single station queueing system with several customer classes. Each customer class has its own arrival process. The total service requirement of each customer is divided into a (possibly) random number of service quanta, where the distribution of each quantum may depend on the customer's class and the other quanta of that customer. The service discipline is round-robin, with random quanta.We prove a heavy traffic limit theorem for the above system which states that as the traffic intensity approaches unity, properly normalized sequences of queue length and sojourn time processes converge weakly to one-dimensional reflected Brownian motion.

Storage-Limited Queues in Heavy Traffic

1991 ◽  
Vol 5 (4) ◽  
pp. 499-522 ◽  
Author(s):  
E. G. Coffman ◽  
A. A. Pukhalskii ◽  
M. I. Reiman
Keyword(s):  
Brownian Motion ◽  
Arrival Time ◽  
Storage Capacity ◽  
Queueing System ◽  
Heavy Traffic ◽  
Capacity Constraints ◽  
Reflected Brownian Motion ◽  
Heavy Traffic Limit ◽  
System 1

This paper models primary computer storage in the context of a general (GI/GI/l) queueing system. Queued items are described by sizes, or storage requirements, as well as by arrival and service times; the sum of the sizes of the items in the system is the occupied storage. Capacity constraints are represented by two different protocols for determining whether an arriving item is admitted to the system: (1) an item is accepted if and only if at its arrival time the currently occupied storage does not exceed a given constantC> 0, and (2) an item is accepted if and only if at its arrival time the occupied storage is at mostC, and the occupied storage plus the item's size is at mostC(l + ε) for some given ε > 0. We prove for both systems that in heavy traffic the occupied storage, suitably normalized, converges weakly to reflected Brownian motion with boundaries at 0 and atC. A distinctive feature of the proof is the characterization of reflected Brownian motion as a limit of unrestricted penalized processes.These results make more plausible an earlier conjecture of the authors, i.e., that one obtains the same heavy traffic limit when the admission rule is: accept an item if and only if at its arrival time the occupied storage plus the item's size is no greater thanC.

Heavy traffic limit theorems for a queueing system in which customers join the shortest line

1989 ◽  
Vol 21 (02) ◽  
pp. 451-469 ◽  
Author(s):  
Zhang Hanqin ◽  
Wang Rongxin
Keyword(s):  
Stochastic Processes ◽  
Central Limit ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Functional Central Limit Theorems ◽  
Service Channels ◽  
Functional Central Limit

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (1) ◽  
pp. 169-183 ◽  
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

An adaptive multistage queueing system

1986 ◽  
Vol 23 (2) ◽  
pp. 495-503
Author(s):  
Bruno Viscolani
Keyword(s):  
Random Number ◽  
Queueing System ◽  
Random Variables ◽  
Arrival Process ◽  
Random Demand ◽  
Exponential Random Variables

The system provides each customer with a service made up of a random number of stages in sequence. Arrivals are Poisson, and the stage times are independent exponential random variables. The number of stages of a particular service depends on the customer's random demand and on the arrival process, in a way which is aimed at preventing the queue from growing fast while matching the customer's demand as well as possible.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (01) ◽  
pp. 169-183
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

Heavy traffic limit theorems for a queueing system in which customers join the shortest line

1989 ◽  
Vol 21 (2) ◽  
pp. 451-469 ◽  
Author(s):  
Zhang Hanqin ◽  
Wang Rongxin
Keyword(s):  
Stochastic Processes ◽  
Central Limit ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Functional Central Limit Theorems ◽  
Service Channels ◽  
Functional Central Limit

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

An adaptive multistage queueing system

1986 ◽  
Vol 23 (02) ◽  
pp. 495-503
Author(s):  
Bruno Viscolani
Keyword(s):  
Random Number ◽  
Queueing System ◽  
Random Variables ◽  
Arrival Process ◽  
Random Demand ◽  
Exponential Random Variables

The system provides each customer with a service made up of a random number of stages in sequence. Arrivals are Poisson, and the stage times are independent exponential random variables. The number of stages of a particular service depends on the customer's random demand and on the arrival process, in a way which is aimed at preventing the queue from growing fast while matching the customer's demand as well as possible.

Diffusion approximations for queues with server vacations

1990 ◽  
Vol 22 (3) ◽  
pp. 706-729 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Heavy Traffic ◽  
Service Discipline ◽  
Single Server ◽  
Traffic Intensity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Reflecting Brownian Motion ◽  
Heavy Traffic Limit

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

Diffusion approximations for queues with server vacations

1990 ◽  
Vol 22 (03) ◽  
pp. 706-729 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Heavy Traffic ◽  
Service Discipline ◽  
Single Server ◽  
Traffic Intensity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Reflecting Brownian Motion ◽  
Heavy Traffic Limit

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

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