scholarly journals M/M/1 Model With Unreliable Service and a Working Vacation

2019 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Stationary Distribution ◽  
Waiting Time ◽  
Queue Length ◽  
Random Variables ◽  
Independent Random Variables ◽  
Working Vacation ◽  
Future Work ◽  
Special Case

We derive an explicit closed form of the stationary distribution of an M/M/1 queue with unreliable service and a working vacation. We also show that the work in (Patterson & Korzeniowski, 2018) can be obtained as a special case of this model. Future work remains to be done; specifically, it may be possible to use the explicit stationary distribution given here to decompose the queue length into the sum of independent random variables. Consequently, it may then be possible to utilize Little’s Law (Little, 1961) to decompose the customer waiting time as well.

scholarly journals Decomposition of M/M/1 With Unreliable Service and a Working Vacation

2020 ◽  
Vol 9 (1) ◽  
pp. 63
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Failure Rate ◽  
Waiting Time ◽  
Queue Length ◽  
Service Failure ◽  
Working Vacation ◽  
Stationary Queue Length ◽  
Stationary Waiting Time ◽  
Relationship Of ◽  
The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (01) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (1) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals Dirichlet and Quasi-Bernoulli Laws for Perpetuities

2014 ◽  
Vol 51 (2) ◽  
pp. 400-416 ◽  
Author(s):  
Paweł Hitczenko ◽  
Gérard Letac
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Bernoulli Distribution ◽  
New Distribution

Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D(a0, …, ad), Pr(B = (0, …, 0, 1, 0, …, 0)) = ai / a with a = ∑i=0dai, and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution Bk(a0, …, ad) with k an integer such that the above result holds when B follows Bk(a0, …, ad) and when Y ~ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a0 = · · · = ad = 1.

scholarly journals HEAVY-TRAFFIC ANALYSIS OF A NON-PREEMPTIVE MULTI-CLASS QUEUE WITH RELATIVE PRIORITIES

2015 ◽  
Vol 29 (2) ◽  
pp. 153-180 ◽  
Author(s):  
A. Izagirre ◽  
I.M. Verloop ◽  
U. Ayesta
Keyword(s):  
Steady State ◽  
State Space ◽  
Waiting Time ◽  
Queue Length ◽  
Heavy Traffic ◽  
Random Variables ◽  
Random Variable ◽  
Traffic Analysis ◽  
Service Requirement ◽  
Heavy Traffic Analysis

We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue-length vector in the heavy-traffic regime, that is, in the limit the scaled queue-length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. We observe that the scaled queue length reduces as classes with smaller mean service requirement obtain relatively larger weights. We finally show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (01) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals Discrete Scan Statistics Generated by Exchangeable Binary Trials

2010 ◽  
Vol 47 (4) ◽  
pp. 1084-1092 ◽  
Author(s):  
Serkan Eryilmaz
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Random Variable ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Binary Trials ◽  
Special Case ◽  
Independent Identically Distributed

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (03) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

Sign in / Sign up

Export Citation Format

Share Document