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.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (02) ◽  
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 servers n grows without a bound, the utilization approaches 1 from below at the rate Assuming 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.

Heavy traffic asymptotics of the queue length in the GI/M/l – K queue

1996 ◽  
Vol 7 (5) ◽  
pp. 519-543 ◽  
Author(s):  
Yongzhi Yang ◽  
Charles Knessl
Keyword(s):  
Queue Length ◽  
Perturbation Methods ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Asymptotic Approximations ◽  
Queue Length Distribution ◽  
Singular Perturbation Methods ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

We consider the GI/M/1 – K queue which has a capacity of K customers. Using singular perturbation methods, we construct asymptotic approximations to the stationary queue length distribution. We assume that K is large and treat several different parameter regimes. Extensive numerical comparisons are used to show the quality of the proposed approximations.

scholarly journals Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

2018 ◽  
Vol 189 ◽  
pp. 02006 ◽  
Author(s):  
S K Koh ◽  
C H Chin ◽  
Y F Tan ◽  
L E Teoh ◽  
A H Pooi ◽  
...  
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Single Server Queue ◽  
Negative Customers ◽  
Stationary Queue Length ◽  
Server Queue

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

scholarly journals Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates

2018 ◽  
Vol 7 (2.15) ◽  
pp. 76
Author(s):  
Koh Siew Khew ◽  
Chin Ching Herny ◽  
Tan Yi Fei ◽  
Pooi Ah Hin ◽  
Goh Yong Kheng ◽  
...  
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Cumulative Distribution ◽  
Interarrival Time ◽  
Single Server ◽  
Idle Period ◽  
Queue Length Distribution ◽  
Breakdown Rates ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be 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.

Traffic light queues as a generalization to queueing theory

1971 ◽  
Vol 8 (3) ◽  
pp. 480-493 ◽  
Author(s):  
Hisashi Mine ◽  
Katsuhisa Ohno
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Traffic Light ◽  
Queue Length Distribution ◽  
Mean Delay ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

scholarly journals Joint queue length distribution of multi-class, single-server queues with preemptive priorities

2015 ◽  
Vol 81 (4) ◽  
pp. 379-395 ◽  
Author(s):  
Andrei Sleptchenko ◽  
Jori Selen ◽  
Ivo Adan ◽  
Geert-Jan van Houtum
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Single Server ◽  
Single Server Queues ◽  
Queue Length Distribution

scholarly journals The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

1998 ◽  
Vol 40 (2) ◽  
pp. 207-221 ◽  
Author(s):  
Yang Woo Shin ◽  
Chareles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Server Vacation ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

scholarly journals Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
Fumio Ishizaki ◽  
Naoto Miyoshi
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Sample Path ◽  
General Setting ◽  
Single Server ◽  
Sample Path Analysis ◽  
Stochastic Version ◽  
Queue Length Distribution ◽  
Probabilistic Setting

In the previous work, the authors have considered a discrete-time queueing system and they have established that, under some assumptions, the stationary queue length distribution for the system with capacity K1 is completely expressed in terms of the stationary distribution for the system with capacity K0 (>K1). In this paper, we study a sample-path version of this problem in more general setting, where neither stationarity nor ergodicity is assumed. We establish that, under some assumptions, the empirical queue length distribution (along through a sample path) for the system with capacity K1 is completely expressed only in terms of the quantities concerning the corresponding system with capacity K0 (>K1). Further, we consider a probabilistic setting where the assumptions are satisfied with probability one, and under the probabilistic setting, we obtain a stochastic version of our main result. The stochastic version is considered as a generalization of the author's previous result, because the probabilistic assumptions are less restrictive.

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