Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (04) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (4) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

The busy cycle of the reflected superposition of Brownian motion and a compound Poisson process

2001 ◽  
Vol 38 (1) ◽  
pp. 255-261 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Closed Form ◽  
Queueing System ◽  
Heavy Traffic ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Maximum Workload ◽  
Busy Cycle

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

The busy cycle of the reflected superposition of Brownian motion and a compound Poisson process

2001 ◽  
Vol 38 (01) ◽  
pp. 255-261
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Closed Form ◽  
Queueing System ◽  
Heavy Traffic ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Maximum Workload ◽  
Busy Cycle

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

INDIVIDUAL EQUILIBRIUM DYNAMIC ROUTING IN A MULTIPLE SERVER RETRIAL QUEUE

2000 ◽  
Vol 14 (1) ◽  
pp. 9-26 ◽  
Author(s):  
Anthony C. Brooms
Keyword(s):  
Nash Equilibrium ◽  
Poisson Process ◽  
Sojourn Time ◽  
Service System ◽  
Transmission Rate ◽  
Retrial Queue ◽  
Dynamic Learning ◽  
Arrival Times ◽  
The Public ◽  
Expected Sojourn Time

Customers arrive sequentially to a service system where the arrival times form a Poisson process of rate λ. The system offers a choice between a private channel and a public set of channels. The transmission rate at each of the public channels is faster than that of the private one; however, if all of the public channels are occupied, then a customer who commits itself to using one of them attempts to connect after exponential periods of time with mean μ−1. Once connection to a public channel has been made, service is completed after an exponential period of time, with mean ν−1. Each customer chooses one of the two service options, basing its decision on the number of busy channels and reapplying customers, with the aim of minimizing its own expected sojourn time. The best action for an individual customer depends on the actions taken by subsequent arriving customers. We establish the existence of a unique symmetric Nash equilibrium policy and show that its structure is characterized by a set of threshold-type strategies; we discuss the relevance of this concept in the context of a dynamic learning scenario.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (04) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

scholarly journals Hubungan Antara Brownian Motion (The Winner Process) dan Surplus Process

2012 ◽  
Vol 12 (1) ◽  
pp. 47
Author(s):  
Tohap Manurung
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Wiener Process ◽  
Compound Poisson Process ◽  
Risk Process ◽  
Actuarial Science ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
Surplus Process ◽  
The Standard Model

HUBUNGAN ANTARA BROWNIAN MOTION (THE WIENER PROCESS) DAN SURPLUS PROCESS ABSTRAK Suatu analisis model continous-time menjadi cakupan yang akan dibahas dalam tulisan ini. Dengan demikian pengenalan proses stochastic akan sangat berperan. Dua proses akan di analisis yaitu proses compound Poisson dan Brownian motion. Proses compound Poisson sudah menjadi model standard untuk Ruin analysis dalam ilmu aktuaria. Sementara Brownian motion sangat berguna dalam teori keuangan modern dan juga dapat digunakan sebagai approksimasi untuk proses compound Poisson. Hal penting dalam tulisan ini adalah menujukkan bagaimana surplus process berdasarkan proses resiko compound Poisson dihubungkan dengan Brownian motion with Drift Process. Kata kunci: Brownian motion with Drift process, proses surplus, compound Poisson   RELATIONSHIP  BETWEEN  BROWNIAN MOTION (THE WIENER PROCESS) AND THE SURPLUS PROCESS ABSTRACT An analysis of continous-time models is covered in this paper. Thus, this requires an introduction to stochastic processes. Two processes are analyzed: the compound Poisson process and Brownian motion. The compound Poisson process has been the standard model for ruin analysis in actuarial science, while Brownian motion has found considerable use in modern financial theory and also can be used as an approximation to the compound Pisson process. The important thing is to show how the surplus process based on compound poisson risk process is related to Brownian motion with drift process. Keywords: Brownian motion with drift process, surplus process, compound Poisson

scholarly journals Single retrial queues with service option on arrival

1997 ◽  
Vol 1 (1) ◽  
pp. 5-12 ◽  
Author(s):  
K. Farahmand ◽  
N. Cooke
Keyword(s):  
Poisson Process ◽  
Queueing System ◽  
Service System ◽  
Queueing Systems ◽  
The State ◽  
Retrial Queues ◽  
General Distribution ◽  
Fixed Rate ◽  
Service Option ◽  
Request Service

We analyze a queueing system in which customers can call in to request service. A proportion, say 1−p of them on their arrival test the availability of the server. If the server is free the customer enters service immediately. Otherwise, if the service system is occupied, the customer joins a source of unsatisfied customers called the orbit. The remaining p proportion of the initial customers enter the orbit directly, without examining the state of the server. We consider two models characterized by the discipline governing the order of re-requests for service from the orbit. First, all the customers from the orbit apply at a fixed rate. Secondly, customers from the orbit are discouraged and reduce their rate of demand as more customers join the orbit. The arrival at and the demands from the orbit are both assumed to be according to the Poisson Process. However, the service times for both primary customers and customers from the orbit are assumed to have a general distribution. We calculate several characteristic quantities of these queueing systems.

scholarly journals Bounds for expected supremum of fractional Brownian motion with drift

2021 ◽  
Vol 58 (2) ◽  
pp. 411-427
Author(s):  
Krzysztof Bisewski ◽  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Fractional Brownian Motion ◽  
Upper Bound ◽  
Numerical Procedure ◽  
Upper And Lower Bounds ◽  
Brownian Motion With Drift ◽  
Similar Shape ◽  
The Mean ◽  
Mean Variance

AbstractWe provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$ , with $B_H(\!\cdot\!)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$ . We find bounds in (semi-) closed form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$ , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$ , the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$ , $H\in(0,\frac{1}{2}]$ , which is tight around $H=\frac{1}{2}$ .

scholarly journals Queueing systems for multiple FBM-based traffic models

2005 ◽  
Vol 46 (3) ◽  
pp. 361-377 ◽  
Author(s):  
Mihaela T. Matache ◽  
Valentin Matache
Keyword(s):  
Brownian Motion ◽  
Upper Bound ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Systems ◽  
Traffic Model ◽  
Queue Size ◽  
Buffer Allocation ◽  
Traffic Models ◽  
Overflow Probability

AbstractA multiple fractional Brownian motion (FBM)-based traffic model is considered. Various lower bounds for the overflow probability of the associated queueing system are obtained. Based on a probabilistic bound for the busy period of an ATM queueing system associated with a multiple FBM-based input traffic, a minimal dynamic buffer allocation function (DBAF) is obtained and a DBAF-allocation algorithm is designed. The purpose is to create an upper bound for the queueing system associated with the traffic. This upper bound, called a DBAF, is a function of time, dynamically bouncing with the traffic. An envelope process associated with the multiple FBM-based traffic model is introduced and used to estimate the queue size of the queueing system associated with that traffic model.

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