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 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 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 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.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

2019 ◽  
Vol 53 (2) ◽  
pp. 367-387
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Renewal Theory ◽  
Function Technique ◽  
Single Server ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

On a single server queue with negative arrivals and request repeated

1999 ◽  
Vol 36 (3) ◽  
pp. 907-918 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
System State ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue ◽  
Regenerative Approach ◽  
Negative Arrivals

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

Large deviations and overflow probabilities for the general single-server queue, with applications

1995 ◽  
Vol 118 (2) ◽  
pp. 363-374 ◽  
Author(s):  
N. G. Duffield ◽  
Neil O'connell
Keyword(s):  
Queue Length ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
General Context ◽  
Scaling Functions ◽  
Single Server ◽  
Tail Probabilities ◽  
Queue Length Distribution ◽  
Image Position

AbstractWe consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (at, vt, t∈R+) and a rate function I such that if (Wt, t∈R+) denotes the workload process, thenon the continuity set of I. In the case that at = vt = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt[8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = supt≥0Wt) decays exponentially:and the decay rate δ is directly related to the rate function I. We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queue-length distribution has an exponential tail only if limt→∞at/vt is finite and strictly positive; otherwise, provided our conditions are satisfied, the tail probabilities decay like

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. 

On a single server queue with negative arrivals and request repeated

1999 ◽  
Vol 36 (03) ◽  
pp. 907-918 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
System State ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue ◽  
Regenerative Approach ◽  
Negative Arrivals

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

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