Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (03) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium

1968 ◽  
Vol 5 (03) ◽  
pp. 579-590 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Fixed Amount ◽  
Deterministic Theory ◽  
Return To Equilibrium ◽  
Maximum Value ◽  
Equilibrium Distributions ◽  
Truncated Normal

During a rush hour, the arrival rateλ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rateμ, and then decrease again. In Part I it was shown that, afterλ(t) has passedμ, the expected queueE{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power ofa(t) =dλ(t)/dtevaluated at timet= 0 whenλ(t) =μ.The maximum ofE{X(t)}, therefore, occurs whenλ(t) again is equal toμat timet1as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L(providedt1is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L+L1) larger than predicted by deterministic theory whereL, is proportional to the (–1/3) power ofa(t1). We also investigate the distribution ofX(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case whereλ(t) is quadratic int.

Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium

1968 ◽  
Vol 5 (3) ◽  
pp. 579-590 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Fixed Amount ◽  
Deterministic Theory ◽  
Return To Equilibrium ◽  
Maximum Value ◽  
Equilibrium Distributions ◽  
Truncated Normal

During a rush hour, the arrival rate λ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate μ, and then decrease again. In Part I it was shown that, after λ(t) has passed μ, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power of a(t) = dλ(t)/dt evaluated at time t = 0 when λ(t) = μ. The maximum of E{X(t)}, therefore, occurs when λ(t) again is equal to μ at time t1 as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L (provided t1 is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L + L1) larger than predicted by deterministic theory where L, is proportional to the (–1/3) power of a(t1). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where λ(t) is quadratic in t.

Exponential upper bounds via martingales for multiplexers with Markovian arrivals

1994 ◽  
Vol 31 (4) ◽  
pp. 1049-1060 ◽  
Author(s):  
E. Buffet ◽  
N. G. Duffield
Keyword(s):  
Markov Processes ◽  
Queue Length ◽  
Heavy Traffic ◽  
Upper Bounds ◽  
Service Rate ◽  
Service Requirement ◽  
Atm Multiplexer ◽  
The Mean ◽  
Analytic Expression ◽  
Mean Queue Length

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {0, 1}, with integral service rate. The bound is of the form [queue length for any where c < 1 and y > 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations.

Exponential upper bounds via martingales for multiplexers with Markovian arrivals

1994 ◽  
Vol 31 (04) ◽  
pp. 1049-1060 ◽  
Author(s):  
E. Buffet ◽  
N. G. Duffield
Keyword(s):  
Markov Processes ◽  
Queue Length ◽  
Heavy Traffic ◽  
Upper Bounds ◽  
Service Rate ◽  
Service Requirement ◽  
Atm Multiplexer ◽  
The Mean ◽  
Analytic Expression ◽  
Mean Queue Length

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {0, 1}, with integral service rate. The bound is of the form [queue length for any where c &lt; 1 and y &gt; 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations.

Social optimization in M/M/1 queue with working vacation and N-policy

2018 ◽  
Vol 52 (2) ◽  
pp. 439-452 ◽  
Author(s):  
Qing-Qing Ma ◽  
Ji-Hong Li ◽  
Wei-Qi Liu
Keyword(s):  
Social Welfare ◽  
Queue Length ◽  
Service Rate ◽  
Working Vacation ◽  
Stationary Probability Distribution ◽  
The Social ◽  
The Mean ◽  
Working Vacations ◽  
Mean Queue Length ◽  
The Impact

This paper deals with the N-policy M/M/1 queueing system with working vacations. Once the system becomes empty, the server begins a working vacation and works at a lower service rate. The server resumes regular service when there are N or more customers in the system. By solving the balance equations, the stationary probability distribution and the mean queue length under observable and unobservable cases are obtained. Based on the reward-cost structure and the theory of Markov process, the social welfare function is constructed. Finally, the impact of several parameters and information levels on the mean queue length and social welfare is illustrated via numerical examples, comparison work shows that queues with working vacations(WV) and N-policy have advantage in controlling the queue length and improving the social welfare.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

scholarly journals Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

2020 ◽  
pp. 119-127
Author(s):  
Orimoloye Segun Michael
Keyword(s):  
Health Care ◽  
Service Delivery ◽  
Queuing Theory ◽  
Health Care Service ◽  
Health Centre ◽  
Arrival Rate ◽  
Queuing System ◽  
Service Rate ◽  
Suitable Model ◽  
The Mean

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (01) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λ i denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μ j the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

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