Busy Period Distribution Research Articles

First-passage-time and busy-period distributions of discrete-time Markovian queues:Geom(n)/Geom(n)/1/N

This paper gives simple explicit solutions of various first-passage-time distributions for a general class of discrete-time queueing models under arbitrary initial conditions, state-dependent transition probabilities and the finite waiting room. Explicit closed form expressions are obtained in terms of roots. These expressions are then used to get numerical as well as graphical results. Explicit closed-form expressions are also deduced for the continuous-time models including the busy-period distributions. The analysis is then extended to cover the case of two absorbing states.

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.

Solving probability transform functional equations for numerical inversion

Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busy-period distribution in the M/G/1 queue. In this paper we provide conditions for iterative methods to converge for complex arguments. Moreover, we show that stochastic monotonicity properties can provide useful bounds.

On the average idle time and average queue length estimates in an M/M/1 queue

We study the bias in the means of average idle time and average queue length estimates, over the interval [0, t], in a transient M/M/1 queue. We develop some formulas, in terms of the busy period distribution, that show that the bias is O( 1 t ) . These formulas yield simple bounds for the means of the transient estimates. The bound for the bias in the mean of the average idle time estimate is also shown to be an excellent approximation.

Transient and busy period analysis of the GI G/1 Queue as a Hilbert factorization problem

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

Transient and busy period analysis of the GIG/1 Queue as a Hilbert factorization problem

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

A new approach to the distribution of the duration of the busy period for a G/G/l queueing system

AbstractFor a G/G/l queueing system let Xt be the number of customers present at time t and Yt(Zt) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let τl be the time until the number of customers in the sustem is reduced from j to j – l, given that X0 = j ≧ l, Y0 = y, Z0 = z. For the joint distribution of τl and Yτl and the Laplace transforms of the τl intergral equations are derived. Under slight conditions these integral equations have unique solutions which can be determined by standard methods. Our results offer a method for calculating the busy period distribution which is completely different from the usual fluctuatuion theoretic approach.

On the busy period distribution of the M/G/2 queueing system

Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.

Performance Analysis Of An Integrated Services Switch

AbstractThe queueing perfonnance of a single server packet-switched system serving two priority classes is analyzed. High priority (HP) packets are assumed to be correlated while low priority (LP) inputs are assumed to obey a Poisson distribution. The system is modeled as a discrete-time single server system and the analysis of the priority service discipline is formulated using an alternating work/vacation model. The transition probabilities of the work/vacation model are manifestations of the correlations within the HP sources. Generating functions (GF) for the queue length, busy period, and packet delay distribution are obtained. It is shown that the correlation present in the HP traffic exerts a significant effect on the queueing performance ofthe LP traffic.

M/G/1/N Queue with vacation time and limited service discipline

A variant of an M/ G/1 queuing model with finite waiting room is studied in this paper. In this system, every busy period of the server at the queue is followed by the execution of additional tasks. The time spent by the server to perform these tasks is called a vacation time away from the queue. The server will begin a vacation from the queue if either the queue has been emptied or M customers have been served during the visit. The embedded Markov chain approach is used to obtain the steady state queue length distribution. The Laplace-Stieltjes transforms of the busy period and cycle time distributions are given. Using a combination of the supplementary variables and sample biasing techniques, the waiting time distribution, blocking probability and general queue length distribution are derived. Finally the results for the case of infinite waiting places are provided.

Haight's distribution and busy periods

Haight's distribution as introduced by Letac and Seshadri is related to busy period distributions and generalised to a semigroup of infinitely divisible distributions.

Transient analysis of a m/m/r machine interference model

A multi–operative Markovian machine interference model with arbitrary inital number of breakdowns is studied by taking mean breakdown of each machine as λ-1 and the mean repairtime of each machine as μ-1. The transient state probabilities of number of breakdowns and initial busy period distributions of a given number of operatives have been obtained in closed form using Laplace transform and matrix theory. The efficacy of the method is tested by computing the state probabilities and measures of effectiveness such as machine availability function and operating efficiency function for various values of traffic intensity ρ=λ/μ.

Busy period distribution in state-dependent queues

We consider a state-dependent M/M/1 queue in which the arrival rate is a function of the instantaneous unfinished work (work backlog) in the system, and the customer's exponential service time distribution is allowed to depend on the unfinished work in the system at the instant that customer arrived. We obtain asymptotic approximations to both the busy period distributions as well as the residual busy period distribution. Our approximations are valid for systems with a rapid arrival rate and small mean service times.

Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed. For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service. When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

A queueing model with bonus service for certain customers

A queueing model is introduced in which the management has a policy, because of economic reasons, of not operating the service counter unless a certain number, R + 1, of customers are available during each busy period. Thus, the first R customers who arrive must wait until the service counter is opened. Such a policy may cause the management to provide or render additional services to the first R customers. Assuming Poisson arrivals and that both regular and additional services follow exponential distributions, explicit expressions are derived for the stationary queue length and busy period distributions and their expected values. In the special case where R = 1, an explicit expression is presented for the stationary distribution of the waiting time.

On the Mx/ G/1 queue with vacation time

We consider an M x / G/1 queueing system in which the server takes a vacation each time that the system, becomes empty. Using supplementary variables, we derive the general queue length distribution at an arbitrary time. We also obtain the waiting time and busy period distributions.

On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines

The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines

The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

PH/M/S/K型待ち行列の過渡現象

We analyze the transient behavior of the PH/M/S/K queue. Emphasis is placed on the overflow process, busy period distribution, idle period distribution, accepted arrival process, departure process and the transition probabilities of the system states. All results are represented by the addition and multiplication of the matrices whose dimensions are at most equal to the number of phases for the inter-arrival time distribution.