Diffusion approximation to a queueing system with time-dependent arrival and service rates

1995 ◽  
Vol 19 (1-2) ◽  
pp. 41-62 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Amelia G. Nobile
Keyword(s):  
Diffusion Approximation ◽  
Queueing System ◽  
Time Dependent ◽  
Service Rates

Time-dependent solution and optimal control of a bulk service queue

1997 ◽  
Vol 34 (01) ◽  
pp. 258-266
Author(s):  
Shokri Z. Selim
Keyword(s):  
Queueing System ◽  
Maximum Size ◽  
Time Dependent ◽  
Optimal Service ◽  
Bulk Service ◽  
Service Rates ◽  
Finite Time Horizon ◽  
Number Of Customers ◽  
Time Dependent Solution ◽  
Dependent Probability

We consider the queueing system denoted by M/MN /1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

Time-dependent solution and optimal control of a bulk service queue

1997 ◽  
Vol 34 (1) ◽  
pp. 258-266 ◽  
Author(s):  
Shokri Z. Selim
Keyword(s):  
Queueing System ◽  
Maximum Size ◽  
Time Dependent ◽  
Optimal Service ◽  
Bulk Service ◽  
Service Rates ◽  
Finite Time Horizon ◽  
Number Of Customers ◽  
Time Dependent Solution ◽  
Dependent Probability

We consider the queueing system denoted by M/MN/1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

Noise in Integrate-and-Fire Neurons: From Stochastic Input to Escape Rates

2000 ◽  
Vol 12 (2) ◽  
pp. 367-384 ◽  
Author(s):  
Hans E. Plesser ◽  
Wulfram Gerstner
Keyword(s):  
Membrane Potential ◽  
Diffusion Approximation ◽  
Hazard Function ◽  
Time Dependent ◽  
Integrate And Fire ◽  
Probability Current ◽  
Periodic Input ◽  
Stochastic Input ◽  
Free Membrane ◽  
Noise Models

We analyze the effect of noise in integrate-and-fire neurons driven by time-dependent input and compare the diffusion approximation for the membrane potential to escape noise. It is shown that for time-dependent subthreshold input, diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the (noise-free) membrane voltage and threshold. The approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage. Stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics.

Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

2021 ◽  
Author(s):  
Gregor Selinka ◽  
Raik Stolletz ◽  
Thomas I. Maindl
Keyword(s):  
Performance Measures ◽  
Real World ◽  
Queueing System ◽  
Time Dependent ◽  
Interval Length ◽  
Arrival Process ◽  
Real World Data ◽  
World Data ◽  
Extant Literature ◽  
Approximation Quality

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline

1988 ◽  
Vol 25 (2) ◽  
pp. 404-412 ◽  
Author(s):  
Julian Keilson ◽  
Ravi Ramaswamy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Time Dependent ◽  
Service Discipline ◽  
Time Process ◽  
State Space Methods ◽  
Ergodic Distribution ◽  
Vacation Model ◽  
Primary Service ◽  
Decomposition Theorems

The vacation model studied is an M/G/1 queueing system in which the server attends iteratively to ‘secondary' or ‘vacation' tasks at ‘primary' service completion epochs, when the primary queue is exhausted. The time-dependent and steady-state distributions of the backlog process [6] are obtained via their Laplace transforms. With this as a stepping stone, the ergodic distribution of the depletion time [5] is obtained. Two decomposition theorems that are somewhat different in character from those available in the literature [2] are demonstrated. State space methods and simple renewal-theoretic tools are employed.

Bias optimal admission control policies for a multiclass nonstationary queueing system

2002 ◽  
Vol 39 (01) ◽  
pp. 20-37 ◽  
Author(s):  
Mark E. Lewis ◽  
Hayriye Ayhan ◽  
Robert D. Foley
Keyword(s):  
Optimal Policy ◽  
Queueing System ◽  
Sufficient Conditions ◽  
System Capacity ◽  
Average Reward ◽  
Finite Capacity ◽  
Long Run ◽  
Optimal Policies ◽  
Service Rates ◽  
Long Run Average Reward

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

scholarly journals Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 979
Author(s):  
Messaoud Bounkhel ◽  
Lotfi Tadj ◽  
Ramdane Hedjar
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Threshold Level ◽  
Tail Probability ◽  
Probability Vector ◽  
Fixed Size ◽  
Markovian Queue ◽  
Entropy Principle ◽  
Service Rates ◽  
Good Agreement

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

On some time-non-homogeneous diffusion approximations to queueing systems

1987 ◽  
Vol 19 (4) ◽  
pp. 974-994 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Approximation ◽  
Busy Period ◽  
Queueing Systems ◽  
Time Dependent ◽  
Single Server ◽  
Time Varying ◽  
Diffusion Approximations ◽  
Linear Drift ◽  
Transition Functions ◽  
Fcfs Discipline

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

Sign in / Sign up

Export Citation Format

Share Document