On Two-Level State-Dependent Routing Polling Systems with Mixed Service

Based on priority differentiation and efficiency of the system, we consider anN+1queues’ single-server two-level polling system which consists of one key queue andNnormal queues. The novel contribution of the present paper is that we consider that the server just polls active queues with customers waiting in the queue. Furthermore, key queue is served with exhaustive service and normal queues are served with 1-limited service in a parallel scheduling. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we derive the explicit closed-form expressions for the mean waiting time. Numerical examples demonstrate that theoretical and simulation results are identical and the new system is efficient both at key queue and normal queues.

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A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964816000528 ◽

2017 ◽

Vol 31 (2) ◽

pp. 139-179 ◽

Cited By ~ 18

Author(s):

Ioannis Dimitriou

Keyword(s):

Performance Metrics ◽

Kernel Method ◽

Length Distribution ◽

Probability Generating Function ◽

Joint Probability ◽

Joint Probability Distribution ◽

Single Server ◽

Method Performance ◽

Service Station ◽

Queue Length Distribution

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

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

RAIRO - Operations Research ◽

10.1051/ro/2017027 ◽

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.

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Queues with moving average service times

Journal of Applied Probability ◽

10.1017/s0021900200025547 ◽

1967 ◽

Vol 4 (03) ◽

pp. 553-570 ◽

Cited By ~ 1

Author(s):

C. Pearce

Keyword(s):

Service Time ◽

Moving Average ◽

Length Distribution ◽

Transient State ◽

Single Server ◽

Queue Length Distribution ◽

Mean And Variance ◽

Poisson Arrivals ◽

The Mean ◽

Service Times

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

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Queues with moving average service times

Journal of Applied Probability ◽

10.2307/3212221 ◽

1967 ◽

Vol 4 (3) ◽

pp. 553-570 ◽

Cited By ~ 4

Author(s):

C. Pearce

Keyword(s):

Service Time ◽

Moving Average ◽

Length Distribution ◽

Transient State ◽

Single Server ◽

Queue Length Distribution ◽

Mean And Variance ◽

Poisson Arrivals ◽

The Mean ◽

Service Times

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New Approach for Finding Basic Performance Measures of Single Server Queue

Journal of Probability and Statistics ◽

10.1155/2014/851738 ◽

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.

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Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

Advances in Applied Probability ◽

10.1239/aap/1214950216 ◽

2008 ◽

Vol 40 (2) ◽

pp. 548-577 ◽

Cited By ~ 14

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.

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M/G/∞ POLLING SYSTEMS WITH RANDOM VISIT TIMES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964808000065 ◽

2007 ◽

Vol 22 (1) ◽

pp. 81-106 ◽

Cited By ~ 7

Author(s):

M. Vlasiou ◽

U. Yechiali

Keyword(s):

Service Time ◽

Probability Generating Function ◽

Expected Value ◽

Polling Systems ◽

Service Time Distribution ◽

Polling System ◽

Stieltjes Transform ◽

Polling Model ◽

Queue Lengths ◽

Number Of Customers

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and the service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, anM/G/∞-type queue. A job that is not completed during a visit will have a new service-time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this article is the first in which anM/G/∞-type polling system is analyzed. For this polling model, we derive the probability generating function and expected value of the queue lengths and the Laplace–Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order isindependentof the number of customers present at the various queues at the start of the cycle.

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The steady-state solution of the M/K2/m queue

Advances in Applied Probability ◽

10.1017/s0001867800035503 ◽

1980 ◽

Vol 12 (03) ◽

pp. 799-823

Author(s):

Per Hokstad

Keyword(s):

Steady State ◽

Queue Length ◽

Simple Formula ◽

Length Distribution ◽

Steady State Solution ◽

Queue Length Distribution ◽

Server Queue ◽

The Mean ◽

Mean Queue Length ◽

The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

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Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽

10.3390/math8111893 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1893

Author(s):

Bara Kim ◽

Jeongsim Kim ◽

Jerim Kim

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Probability Generating Function ◽

Finite Collection ◽

Numerical Inversion ◽

Analytic Method ◽

Waiting Time Distribution ◽

Mean Waiting Time ◽

The Matrix ◽

The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

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The Large Deviations of Estimating Rate Functions

Journal of Applied Probability ◽

10.1017/s0021900200000218 ◽

2005 ◽

Vol 42 (01) ◽

pp. 267-274 ◽

Cited By ~ 3

Author(s):

Ken Duffy ◽

Anthony P. Metcalfe

Keyword(s):

Large Deviation Principle ◽

Large Deviation ◽

Length Distribution ◽

Rate Function ◽

Partial Sums ◽

Single Server ◽

Mixing Condition ◽

Deviation Principle ◽

Queue Length Distribution ◽

Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

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