scholarly journals An approximate mean queue length formula for queueing systems with varying service rate

2021 ◽  
Vol 17 (1) ◽  
pp. 185-204
Author(s):  
Jian Zhang ◽  
◽  
Tony T. Lee ◽  
Tong Ye ◽  
Liang Huang ◽  
...  
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Service Rate ◽  
Mean Queue Length

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.

Use of erlangian distributions for single-server queueing systems

1969 ◽  
Vol 6 (3) ◽  
pp. 584-593 ◽  
Author(s):  
T. C. T. Kotiah ◽  
J. W. Thompson ◽  
W. A. O'N. Waugh
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Single Server ◽  
System A ◽  
Bulk Service ◽  
Mean And Variance ◽  
The Mean ◽  
Mean Queue Length ◽  
Mean Time ◽  
Server System

SummaryThe use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.

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.

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.

Use of erlangian distributions for single-server queueing systems

1969 ◽  
Vol 6 (03) ◽  
pp. 584-593 ◽  
Author(s):  
T. C. T. Kotiah ◽  
J. W. Thompson ◽  
W. A. O'N. Waugh
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Single Server ◽  
System A ◽  
Bulk Service ◽  
Mean And Variance ◽  
The Mean ◽  
Mean Queue Length ◽  
Mean Time ◽  
Server System

Summary The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.

scholarly journals A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Ekaterina Evdokimova ◽  
Sabine Wittevrongel ◽  
Dieter Fiems
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Poisson Processes ◽  
Series Expansions ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
State Space Explosion ◽  
Finite Queues ◽  
Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

Filtering of Markov renewal queues, IV: Flow processes in feedback queues

1985 ◽  
Vol 17 (2) ◽  
pp. 386-407 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Flow Process ◽  
Special Cases ◽  
Flow Processes ◽  
Transition Points ◽  
The Times

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

Sign in / Sign up

Export Citation Format

Share Document