A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

Download Full-text

Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

Advances in Operations Research ◽

10.1155/2016/2456135 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

K. V. Abdul Rasheed ◽

M. Manoharan

Keyword(s):

Queueing System ◽

Queueing Systems ◽

Single Server ◽

Expected Number ◽

Multiple Servers ◽

Special Cases ◽

Service Rates ◽

Expected Waiting Time ◽

Number Of Customers ◽

Steady State Probabilities

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

Download Full-text

Some Time-Dependent Properties of Symmetric M/G/1 Queues

Journal of Applied Probability ◽

10.1017/s0021900200000176 ◽

2005 ◽

Vol 42 (01) ◽

pp. 223-234 ◽

Cited By ~ 1

Author(s):

Offer Kella ◽

Bert Zwart ◽

Onno Boxma

Keyword(s):

Queue Length ◽

Marginal Distribution ◽

Geometric Distribution ◽

Time Dependent ◽

Service Discipline ◽

Processor Sharing ◽

Tail Behavior ◽

Exponential Time ◽

Preemptive Resume ◽

Number Of Customers

We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

Download Full-text

Concavity of the throughput of tandem queueing systems with finite buffer storage space

Advances in Applied Probability ◽

10.1017/s0001867800020024 ◽

1990 ◽

Vol 22 (03) ◽

pp. 764-767 ◽

Cited By ~ 7

Author(s):

Ludolf E. Meester ◽

J. George Shanthikumar

Keyword(s):

Queueing System ◽

Queueing Systems ◽

Sample Path ◽

Concave Function ◽

Time Interval ◽

Single Server ◽

Finite Buffer ◽

Buffer Storage ◽

Unlimited Supply ◽

Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

Download Full-text

On extremal service disciplines in single-stage queueing systems

Journal of Applied Probability ◽

10.1017/s0021900200038869 ◽

1990 ◽

Vol 27 (02) ◽

pp. 409-416 ◽

Cited By ~ 4

Author(s):

Rhonda Righter ◽

J. George Shanthikumar ◽

Genji Yamazaki

Keyword(s):

Service Time ◽

Sojourn Time ◽

Queueing System ◽

Residual Life ◽

Queueing Systems ◽

Mean Residual Life ◽

Service Discipline ◽

Service Time Distribution ◽

Service Disciplines ◽

Number Of Customers

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

Download Full-text

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

Journal of Applied Probability ◽

10.1017/s0021900200038882 ◽

1990 ◽

Vol 27 (02) ◽

pp. 425-432

Author(s):

Hahn-Kyou Rhee ◽

B. D. Sivazlian

Keyword(s):

Steady State ◽

Busy Period ◽

Queueing System ◽

Service Completion ◽

Special Cases ◽

Service Stations ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Number Of Customers ◽

System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

Download Full-text

A solvable model for a finite-capacity queueing system

Journal of Applied Probability ◽

10.2307/3213957 ◽

1985 ◽

Vol 22 (4) ◽

pp. 903-911 ◽

Cited By ~ 14

Author(s):

V. Giorno ◽

C. Negri ◽

A. G. Nobile

Keyword(s):

Queueing System ◽

Waiting Times ◽

Queueing Systems ◽

Solvable Model ◽

Probability Density Functions ◽

Single Server ◽

Finite Capacity ◽

System Service ◽

Transient Probabilities ◽

Number Of Customers

Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

Download Full-text

On extremal service disciplines in single-stage queueing systems

Journal of Applied Probability ◽

10.2307/3214660 ◽

1990 ◽

Vol 27 (2) ◽

pp. 409-416 ◽

Cited By ~ 38

Author(s):

Rhonda Righter ◽

J. George Shanthikumar ◽

Genji Yamazaki

Keyword(s):

Service Time ◽

Sojourn Time ◽

Queueing System ◽

Residual Life ◽

Queueing Systems ◽

Mean Residual Life ◽

Service Discipline ◽

Service Time Distribution ◽

Service Disciplines ◽

Number Of Customers

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

Download Full-text

Concavity of the throughput of tandem queueing systems with finite buffer storage space

Advances in Applied Probability ◽

10.2307/1427472 ◽

1990 ◽

Vol 22 (3) ◽

pp. 764-767 ◽

Cited By ~ 40

Author(s):

Ludolf E. Meester ◽

J. George Shanthikumar

Keyword(s):

Queueing System ◽

Queueing Systems ◽

Sample Path ◽

Concave Function ◽

Time Interval ◽

Single Server ◽

Finite Buffer ◽

Buffer Storage ◽

Unlimited Supply ◽

Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

Download Full-text

Point processes generated by transitions of Markov chains

Advances in Applied Probability ◽

10.2307/1426036 ◽

1973 ◽

Vol 5 (2) ◽

pp. 262-286 ◽

Cited By ~ 21

Author(s):

Mats Rudemo

Keyword(s):

Markov Chain ◽

Point Process ◽

Point Processes ◽

Queueing System ◽

Initial Distribution ◽

Recurrence Time ◽

Asynchronous Sampling ◽

Doubly Stochastic ◽

Special Cases ◽

Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

Download Full-text