scholarly journals Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals

2021 ◽  
Vol 11 (3) ◽  
pp. 264-283
Author(s):  
Xinyun Chen
Keyword(s):  
Moment Generating Function ◽  
General Distribution ◽  
Single Server ◽  
Perfect Sampling ◽  
Sampling Algorithm ◽  
Finite Moment ◽  
Hawkes Processes ◽  
Numerical Tests ◽  
Light Tail ◽  
The Stability

In this paper we develop to our best knowledge the first perfect sampling algorithm for queues with Hawkes input (i.e., single-server queues with Hawkes arrivals and independent and identically distributed service times of general distribution). In addition to the stability condition, we also assume the excitation function of the Hawkes process has a light tail and the service time has finite moment-generating function in the neighborhood of the origin. In this procedure, we also propose a new perfect sampling algorithm for Hawkes processes with improved computational efficiency compared with the existing algorithm. Theoretical analysis and numerical tests on the algorithms’ correctness and efficiency are also included.

ON THE WORKLOAD PROCESS IN A FLUID QUEUE WITH A RESPONSIVE BURSTY INPUT AND SELECTIVE DISCARDING

2003 ◽  
Vol 17 (4) ◽  
pp. 527-543
Author(s):  
Parijat Dube ◽  
Eitan Altman
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Feedback System ◽  
Moment Generating Function ◽  
General Distribution ◽  
Finite Buffer ◽  
Fluid Queue ◽  
The Stability ◽  
The Moment ◽  
Stationary Workload

We analyze a feedback system consisting of a finite buffer fluid queue and a responsive source. The source alternates between silence periods and active periods. At random epochs of times, the source becomes ready to send a burst of fluid. The length of the bursts (length of the active periods) are independent and identically distributed with some general distribution. The queue employs a threshold discarding policy in the sense that only those bursts at whose commencement epoch (the instant at which the source is ready to send) the workload (i.e., the amount of fluid in the buffer) is less than some preset threshold are accepted. If the burst is rejected then the source backs off from sending. We work within the framework of Poisson counter-driven stochastic differential equations and obtain the moment generating function and hence the probability density function of the stationary workload process. We then comment on the stability of this fluid queue. Our explicit characterizations will further provide useful insights and “engineering” guidelines for better network designing.

scholarly journals STABILITY ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

2014 ◽  
Vol 31 (02) ◽  
pp. 1440002 ◽  
Author(s):  
K. AVRACHENKOV ◽  
E. MOROZOV ◽  
R. NEKRASOVA ◽  
B. STEYAERT
Keyword(s):  
Queueing System ◽  
Stability Domain ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Single Orbit ◽  
Transmission Control Protocols ◽  
Regenerative Approach ◽  
The Stability ◽  
Multi Access

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

The M/G/1 queue with negative customers

1996 ◽  
Vol 28 (02) ◽  
pp. 540-566 ◽  
Author(s):  
Peter G. Harrison ◽  
Edwige Pitel
Keyword(s):  
Iterative Solution ◽  
Single Server ◽  
Negative Customers ◽  
Poisson Arrival ◽  
First Case ◽  
Full Study ◽  
Service Times ◽  
The Stability ◽  
The One ◽  
Iterative Solution Method

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (1) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

Stability condition for a single-server retrial queue

1993 ◽  
Vol 25 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Huei-Mei Liang ◽  
V. G. Kulkarni
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Retrial Queue ◽  
Arrival Rate ◽  
Retrial Queues ◽  
Single Server ◽  
Sufficient Stability Condition ◽  
Sufficient Stability ◽  
The Mean ◽  
The Stability

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system. We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ / μ < 1 is a sufficient stability condition for the M/G /1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

Numerical Tests of Tunnel Reinforcing Influences on Zonal Disintegration within Rockmass around Deep Tunnel

2011 ◽  
Vol 261-263 ◽  
pp. 1282-1286
Author(s):  
Yu Jun Zuo ◽  
Yong Bin Zhang ◽  
Shu Cai Li ◽  
Yi Ping Zhang ◽  
Chun Chun Chen
Keyword(s):  
Failure Process ◽  
Vertical Wall ◽  
Three Dimensional ◽  
General Nature ◽  
Zonal Disintegration ◽  
Deep Tunnel ◽  
Tunnel Wall ◽  
Numerical Tests ◽  
Zonal Disintegration Phenomenon ◽  
The Stability

Nnumerical tests on three-dimensional failure process of rock samples containing vertical wall semi-arched tunnel with and without reinforcing are carried out with Mechsoft’s RFPA-Parallel system running on Lenovo 1800 Cluster, reproducing zonal disintegration phenomenon within rockmass around deep tunnels, and then the deep tunnel reinforcing influences on zonal disintegration within rockmass around tunnel is analyzed. Numerical results indicate that deep tunnel reinforcing does not change the general nature to form zonal disintegration phenomenon, but it can improve the stability of the tunnel wall.

scholarly journals On the stability of polling models with multiple servers

1998 ◽  
Vol 35 (04) ◽  
pp. 925-935 ◽  
Author(s):  
D. Down
Keyword(s):  
Stability Conditions ◽  
Single Server ◽  
Fluid Limit ◽  
Polling Models ◽  
Multiple Servers ◽  
The Stability ◽  
Limit Models

The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

scholarly journals Speeding up the FMMR perfect sampling algorithm: A case study revisited

2003 ◽  
Vol 23 (4) ◽  
pp. 434-452
Author(s):  
Robert P. Dobrow ◽  
James Allen Fill
Keyword(s):  
Perfect Sampling ◽  
Sampling Algorithm
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