On queues with periodic inputs

1989 ◽  
Vol 26 (02) ◽  
pp. 381-389 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Limit Theorems ◽  
Single Server ◽  
Single Server Queue ◽  
Input Process ◽  
Poisson Arrival ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Waiting Time Process ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

On queues with periodic inputs

1989 ◽  
Vol 26 (2) ◽  
pp. 381-389 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Limit Theorems ◽  
Single Server ◽  
Single Server Queue ◽  
Input Process ◽  
Poisson Arrival ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Waiting Time Process ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

Assembly-like queues

1973 ◽  
Vol 10 (2) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

Assembly-like queues

1973 ◽  
Vol 10 (02) ◽  
pp. 354-367 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Waiting Time ◽  
Limit Theorems ◽  
Theoretic Model ◽  
Single Server ◽  
Time Process ◽  
Single Server Queue ◽  
Multiple Input ◽  
Server Queue ◽  
Waiting Time Process ◽  
Load Conditions

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

Asymptotic behavior of a multiplexer fed by a long-range dependent process

1999 ◽  
Vol 36 (01) ◽  
pp. 105-118 ◽  
Author(s):  
Zhen Liu ◽  
Philippe Nain ◽  
Don Towsley ◽  
Zhi-Li Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Lognormal Distribution ◽  
Arrival Rate ◽  
Upper Bounds ◽  
Single Server ◽  
Service Capacity ◽  
Single Server Queue ◽  
Input Process ◽  
Server Queue

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ < 1, where ρ is the arrival rate at the buffer.

On stability of queueing networks with job deadlines

2003 ◽  
Vol 40 (2) ◽  
pp. 293-304 ◽  
Author(s):  
Amy R. Ward ◽  
Nicholas Bambos
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Single Server ◽  
Single Server Queue ◽  
Interesting Phenomenon ◽  
Single Node ◽  
Service Disciplines ◽  
Server Queue ◽  
Weakly Stable ◽  
Networks Of Queues

In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

scholarly journals Estimating the Probability of Ruin for Variable Premiums by Simulation

1996 ◽  
Vol 26 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Frédéric Michaud
Keyword(s):  
Sample Path ◽  
Risk Theory ◽  
Single Server ◽  
Single Server Queue ◽  
Storage Process ◽  
Probability Of Ruin ◽  
Surplus Process ◽  
Special Cases ◽  
Server Queue ◽  
Waiting Time Process

AbstractThere is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

Sojourn times in single-server queues by negative customers

1993 ◽  
Vol 30 (4) ◽  
pp. 943-963 ◽  
Author(s):  
P. G. Harrison ◽  
E. Pitel
Keyword(s):  
Laplace Transform ◽  
Single Server ◽  
Processor Sharing ◽  
Single Server Queue ◽  
Negative Customers ◽  
Mean Values ◽  
Poisson Arrival ◽  
Server Queue ◽  
The Laplace Transform ◽  
Time Density

We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.

The total waiting time in a busy period of a stable single-server queue, II

1969 ◽  
Vol 6 (3) ◽  
pp. 565-572 ◽  
Author(s):  
D. J. Daley ◽  
D. R. Jacobs
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Bessel Functions ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Server Queue ◽  
Common Distribution Function

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.

On the single-server queue with non-homogeneous Poisson input and general service time

1964 ◽  
Vol 1 (2) ◽  
pp. 369-384 ◽  
Author(s):  
A. M. Hasofer
Keyword(s):  
Periodic Function ◽  
Waiting Time ◽  
Service Time ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Asymptotically Periodic ◽  
Image Position ◽  
General Service

In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both Po and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained.In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day

On queues in discrete regenerative environments, with application to the second of two queues in series

1979 ◽  
Vol 11 (04) ◽  
pp. 851-869 ◽  
Author(s):  
K. Balagopal
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Single Server ◽  
Single Server Queue ◽  
Light Traffic ◽  
Queuing Model ◽  
Heavy Traffic Limit ◽  
Server Queue ◽  
In Series ◽  
Queues In Series

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn , n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model. In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

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