Steady-state Waiting Time Distribution Research Articles

Calculation of the steady state waiting time distribution in the context of the machine interference problem

This paper proposes an extension of a multinomial model for the machine interference problem, where each of N identical machines randomly requests several different service types. Each request for a service is fulfilled by an operator who can provide only one type of service. The extended model is useful for the case in which there is a time limit for one service type (or more), such that it must be accomplished within a certain time from the moment of the request. A delay in the service above a given time spoils the product and makes it useless for its intended purpose. The model allows calculation of the exact distribution function of the steady state waiting time and total service time (waiting time + service time) for each type of requested service, for the first come first served (FCFS) queue discipline.

Calculation of the steady state waiting time distribution in the context of the machine interference problem

This paper proposes an extension of a multinomial model for the machine interference problem, where each of N identical machines randomly requests several different service types. Each request for a service is fulfilled by an operator who can provide only one type of service. The extended model is useful for the case in which there is a time limit for one service type (or more), such that it must be accomplished within a certain time from the moment of the request. A delay in the service above a given time spoils the product and makes it useless for its intended purpose. The model allows calculation of the exact distribution function of the steady state waiting time and total service time (waiting time + service time) for each type of requested service, for the first come first served (FCFS) queue discipline.

Queues and Risk Processes with Dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in Ref.[13]. In this setting, we obtain the steady-state waiting time distribution, and we show that the classical relation between the steady-state waiting time and workload distributions remains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally, we show that there exist stochastic order relations between the waiting times under various instances of correlation.

A first passage time problem for spectrally positive Lévy processes and its application to a dynamic priority queue

We study a first passage time problem for a class of spectrally positive Lévy processes. By considering the special case where the Lévy process is a compound Poisson process with negative drift, we obtain the Laplace–Stieltjes transform of the steady-state waiting time distribution of low-priority customers in a two-class M/GI/1 queue operating under a dynamic non-preemptive priority discipline. This allows us to observe how the waiting time of customers is affected as the policy parameter varies.

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.

Uniform approximations for the M/G/1 queue with subexponential processing times

This paper studies the asymptotic behavior of the steady-state waiting time, W ?, of the M/G/1 queue with Subexponential processing times for different combinations of traffic intensities and overflow levels. In particular, we provide insights into the regions of large deviations where the so-called heavy-traffic approximation and heavy-tail asymptotic hold. For queues whose service time distribution decays slower than $e^{-\sqrt{t}}$ we identify a third region of asymptotics where neither the heavy-traffic nor the heavy-tail approximations are valid. These results are obtained by deriving approximations for P(W ?>x) that are either uniform in the traffic intensity as the tail value goes to infinity or uniform on the positive axis as the traffic intensity converges to one. Our approach makes clear the connection between the asymptotic behavior of the steady-state waiting time distribution and that of an associated random walk.

Queues with waiting time dependent service

Motivated by service levels in terms of the waiting-time distribution seen, for instance, in call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steady-state waiting-time distributions for both models. The results are illustrated with numerical examples.

Service Competition with General Queueing Facilities

In many service industries, companies compete with each other on the basis of the waiting time their customers' experience, along with the price they charge for their service. A firm's waiting time standard may either be defined in terms of the expected value or a given, for instance 95%, percentile of the steady state waiting time distribution. We investigate how a service industry's competitive behavior depends on the characteristics of the service providers' queueing systems. We provide a unifying approach to investigate various standard single stage systems covering the spectrum from M/M/1 to general G/GI/s systems, along with open Jackson networks to represent multi-stage service systems. Assuming that the capacity cost is proportional with the service rates we refer to its dependence on (i) the firm's demand rate and (ii) the waiting time standard as the capacity cost function. We show that across the above road spectrum of queueing models, the capacity cost function belongs to a specific four parameter class of function, either exactly or as a close approximation. We then characterize how this capacity cost function impacts on the equilibrium behavior in the industry. We give separate treatments to the case where the firms compete in terms of (i) prices (only) (ii) their service level or waiting time standard (only), (iii) simultaneously in terms of both prices and service levels. The firms' demand rates are given by a general system of equations of the prices and waiting time standards in the industry.

Engineering Solution of a Basic Call-Center Model

An algorithm is developed to rapidly compute approximations for all the standard steady-state performance measures in the basic call-center queueing model M/GI/s/r+GI, which has a Poisson arrival process, independent and identically distributed (IID) service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies of call centers indicate that the service-time and abandon-time distributions often are not nearly exponential, so that it is important to go beyond the Markovian M/M/s/r+M special case, but the general service-time and abandon-time distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an appropriate Markovian M/M/s/r+M(n) queueing model, where M(n) denotes state-dependent abandonment rates. After making an additional approximation, steady-state waiting-time distributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the abandonment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.

The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution

By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the first-come first-served discipline are bounded below by tails of Poisson distributions. As a consequence, the tail of the steady-state waiting-time distribution is bounded below by a constant times the sth power of the tail of the service-time stationary-excess distribution. We apply that bound to show that the steady-state waiting-time distribution has a heavy tail (with appropriate definition) whenever the service-time distribution does. We also establish additional results that enable us to nearly capture the full asymptotics in both light and heavy traffic. The difference between the asymptotic behavior in these two regions shows that the actual asymptotic form must be quite complicated.

Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions

O.J. Boxma and J.W. Cohen recently obtained an explicit expression for the M/G/1 steady-state waiting-time distribution for a class of service-time distributions with power tails. We extend their explicit representation from a one-parameter family of service-time distributions to a two-parameter family. The complementary cumulative distribution function (ccdf's) of the service times all have the asymptotic form F c( t)∼ αt −3/2 as t→∞, so that the associated waiting-time ccdf's have asymptotic form W c( t)∼ βt −1/2 as t→∞. Thus the second moment of the service time and the mean of the waiting time are infinite. Our result here also extends our own earlier explicit expression for the M/G/1 steady-state waiting-time distribution when the service-time distribution is an exponential mixture of inverse Gaussian distributions (EMIG). The EMIG distributions form a two-parameter family with ccdf having the asymptotic form F c( t)∼ αt −3/2e − ηt as t→∞. We now show that a variant of our previous argument applies when the service-time ccdf is an undamped EMIG, i.e., with ccdf G c( t)=e ηt F c( t) for F c( t) above, which has the power tail G c( t)∼ αt −3/2 as t→∞. The Boxma–Cohen long-tail service-time distribution is a special case of an undamped EMIG.

Representations for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues

Neuts has shown that a Markov chain of GI/M/1 type has a matrix-geometric steady-state distribution, extending earlier work by Evans [13] and Wallace [31]. Various authors later observed that, under this steady-state distribution, the “level” component is phase type. We give a probabilistic interpretation of this result by constructing, from a sample path of the original process, a Markov chain with the correct distribution of time to absorption. We give a similar construction for the matrix-exponential steady-state distribution of Sengupta's continuous-level chain [27]. We extend Sengupta's analysis of the GI/PH/1 queue to the many-server case and thereby prove that the steady-state waiting time distribution in a GI/PH/c queue is phase type. This leads to a proof of a recent conjecture of Abate, Choudhury and Whitt [1] and gives an alternative approach to Neuts and Takahashi's [20] results on the exponential decay parameter of the waiting-time distribution

Limits and approximations for the M/G/1 LIFO waiting-time distribution

We provide additional descriptions of the steady-state waiting-time distribution in the M/G/1 queue with the last-in firstout (LIFO) service discipline. We establish heavy-traffic limits for both the cumulative distribution function (cdf) and the moments. We develop an approximation for the cdf that is asymptotically correct both as the traffic intensity ρ → 1 for each time t and as t → ∞ for each ρ. We show that in heavy traffic the LIFO moments are related to the FIFO moments by the Catalan numbers. We also develop a new recursive algorithm for computing the moments.

A discrete single server queue with Markovian arrivals and phase type group services

We consider a single‐server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady‐state probability vector is shown to be of (modified) matrix‐geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady‐state probability vector, and some important system performance measures are developed. The steady‐state waiting time distribution is derived explicitly. Some numerical examples are presented.

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x –1 log P(W > x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n –1 log E exp (θSn ) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the formx–1logP(W> x) → –θ∗asx→ ∞forθ∗>0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e.n–1logEexp (θSn) →ψ(θ) asn→ ∞, plus regularity conditions on the decay rate functionψ. The asymptotic decay rateθis the root of the equationψ(θ)=0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

Understanding the Efficiency of Multi-Server Service Systems

In the design and operation of service systems, it is important to determine an appropriate level of server utilization (the proportion of time each server should be working). In a multi-server queue with unlimited waiting space, the appropriate server utilization typically increases as the number of servers (and the arrival rate) increases. We explain this economy of scale and give a rough quantitative characterization. We also show how increased variability in the arrival and service processes tends to reduce server utilization with a given grade of service. As part of this analysis, we develop simple approximations for the mean steady-state waiting time and the full steady-state waiting-time distribution. These approximations exploit an infinite-server approximation for the probability of delay and a single-server approximation for the conditional waiting-time distribution given that waiting occurs. The emphasis is on simple formulas that directly convey understanding.

The discrete-time single-server queue

In this paper we consider the discrete-time single server queueing model with exceptional first service. For this model we cannot define the steady-state waiting-time distribution simply as the limiting distribution of the waiting times, since this limit does not always exist. Instead, we use the Cesaro limit to define the limiting waiting-time distribution. We give an exact relation between the generating functions of the steady-state waiting-time distribution and of the idle-time distribution in the case of general interarrival-time and service-time distributions. Once we have this relation, we can give more explicit results when the generating function of either the interarrival-time distribution or the service-time distribution is rational. We also derive some results on the asymptotic behaviour of the waiting-time distribution.

GI/G/1 Interdeparture time and queue-length distributions via the Laguerre transform

Queue length and interdeparture distributions for GI/G/1 are obtained using the Laguerre function expansion of the waiting time distribution. The expansion of the steady state waiting time distribution is obtained here by solving a small set of linear equations in the Laguerre function expansion coefficients. Examples show the accuracy of the results and illustrate purely numerical techniques for obtaining the necessary expansions of the arrival and service distributions.

A generalized pollaczek-khinchine formula for the gi/gi/1/k queue and its application to approximation

This paper deals with the steady-state waiting time distribution in the GI/GI/1/k {GI/GI/1 with k waiting places) queue. Marshall[8]'s generalized Pollaczek-Khinchine formula is generalized to the GI/GI/1/k model. From the resulting formulas, we obtain approximation formulas for the waiting time distribution and for the probabilities of the system being empty and of customer loss. The quality of those approximations is shown by numerical examples.