Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues

1994 ◽  
Vol 31 (A) ◽  
pp. 251-267
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Random Walks ◽  
Queueing Theory ◽  
Queue Length ◽  
Heuristic Method ◽  
Length Distribution ◽  
Arrival Rate ◽  
Brownian Excursion ◽  
Queue Length Distribution ◽  
Ballot Theorem ◽  
Exit Probabilities

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion. The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues

1994 ◽  
Vol 31 (A) ◽  
pp. 251-267 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Random Walks ◽  
Queueing Theory ◽  
Queue Length ◽  
Heuristic Method ◽  
Length Distribution ◽  
Arrival Rate ◽  
Brownian Excursion ◽  
Queue Length Distribution ◽  
Ballot Theorem ◽  
Exit Probabilities

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

Exploiting Markov chains to infer queue length from transactional data

1992 ◽  
Vol 29 (3) ◽  
pp. 713-732 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Markov Chains ◽  
Queue Length ◽  
Telecommunication Network ◽  
Length Distribution ◽  
Exact Algorithm ◽  
Arrival Rate ◽  
Finite Buffers ◽  
Queue Length Distribution ◽  
Time Estimates ◽  
Poisson Arrivals

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n3) algorithm for busy periods with n customers and an O(n2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.

Exploiting Markov chains to infer queue length from transactional data

1992 ◽  
Vol 29 (03) ◽  
pp. 713-732 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Markov Chains ◽  
Queue Length ◽  
Telecommunication Network ◽  
Length Distribution ◽  
Exact Algorithm ◽  
Arrival Rate ◽  
Finite Buffers ◽  
Queue Length Distribution ◽  
Time Estimates ◽  
Poisson Arrivals

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n 3) algorithm for busy periods with n customers and an O(n 2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.

scholarly journals Performance Analysis of Novel Overload Control with Threshold Mechanism

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Doo Il Choi ◽  
Dae-Eun Lim
Keyword(s):  
Queue Length ◽  
Control Method ◽  
Length Distribution ◽  
Arrival Rate ◽  
Embedded Markov Chain ◽  
Overload Control ◽  
Queue Length Distribution ◽  
Markov Chain Method ◽  
Mean Waiting Time ◽  
Threshold Mechanism

We propose a novel overload control method with hysteresis property; that is, we analyze theM/G/1/Kqueueing system where the service and arrival rates are varied depending on the queue-length. We use two threshold values:L1(≤L2)andL2(≤K). When the queue-length increases by an amount betweenL1andL2, we apply one of the following two strategies to reduce the queue-length, either we decrease the mean service time or we decrease the arrival rate. If the queue-length exceedsL2with one strategy, we apply the other; thus, there are two models that depend on the method that was applied first. We derive the queue-length distribution at departure and at arbitrary epochs using the embedded Markov chain method and the supplementary variable method. We investigate performance measures including the loss probability and mean waiting time using various numerical examples.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

scholarly journals New Approach for Finding Basic Performance Measures of Single Server Queue

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Siew Khew Koh ◽  
Ah Hin Pooi ◽  
Yi Fei Tan
Keyword(s):  
Stationary State ◽  
Queue Length ◽  
Length Distribution ◽  
Cumulative Distribution ◽  
System Capacity ◽  
Interarrival Time ◽  
Single Server ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue

Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

scholarly journals Approximation of The Queue Length Distribution of General Queues

ETRI Journal ◽  
1994 ◽  
Vol 15 (3) ◽  
pp. 35-45 ◽  
Author(s):  
Kyu-Seok Lee ◽  
Hong Shik Park
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Queue Length Distribution

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.

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