A note on Bayesian estimation of traffic intensity in single-server Markovian queues

2017 ◽  
Vol 47 (9) ◽  
pp. 2577-2586 ◽  
Author(s):  
Márcio A. C. Almeida ◽  
Frederico R. B. Cruz
Keyword(s):  
Bayesian Estimation ◽  
Single Server ◽  
Traffic Intensity ◽  
Markovian Queues

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

scholarly journals Equilibrium and Optimal Strategies in M/M/1 Queues with Working Vacations and Vacation Interruptions

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Ruiling Tian ◽  
Linmin Hu ◽  
Xijun Wu
Keyword(s):  
Optimal Strategies ◽  
Cost Structure ◽  
Single Server ◽  
Equilibrium Strategies ◽  
Markovian Queues ◽  
Multiple Working Vacations ◽  
Working Vacations ◽  
System States ◽  
Partially Observable ◽  
Stationary Behavior

We consider the customers equilibrium and socially optimal joining-balking behavior in single-server Markovian queues with multiple working vacations and vacation interruptions. Arriving customers decide whether to join the system or balk, based on a linear reward-cost structure that incorporates their desire for service, as well as their unwillingness for waiting. We consider that the system states are observable, partially observable, and unobservable, respectively. For these cases, we first analyze the stationary behavior of the system and get the equilibrium strategies of the customers and compare them to socially optimal balking strategies numerically.

Generalised birth and death queueing processes: recent results

1977 ◽  
Vol 9 (1) ◽  
pp. 125-140 ◽  
Author(s):  
B. W. Conolly ◽  
J. Chan
Keyword(s):  
Waiting Time ◽  
Queueing Systems ◽  
System Size ◽  
Single Server ◽  
Traffic Intensity ◽  
Stable Regime ◽  
Mean Waiting Time ◽  
Service Rates ◽  
Queueing Processes ◽  
Effective Service

The systems considered are single-server, though the theory has wider application to models of adaptive queueing systems. Arrival and service mechanisms are governed by state (n)-dependent mean arrival and service rates λn and µn. It is assumed that the choice of λn and µn leads to a stable regime. Formulae are sought that provide easy means of computing statistics of effectiveness of systems. A measure of traffic intensity is first defined in terms of ‘effective’ service time and inter-arrival intervals. It is shown that the latter have a renewal type connection with appropriately defined mean effective arrival and service rates λ∗ and µ∗ and that in consequence the ratio λ∗/µ∗ is the traffic intensity, equal moreover to where is the stable probability of an empty system, consistent with other systems. It is also shown that for first come, first served discipline the equivalent of Little's formula holds, where and are the mean waiting time of an arrival and mean system size at an arbitrary epoch. In addition it appears that stable regime output intervals are statistically identical with effective inter-arrival intervals. Symmetrical moment formulae of arbitrary order are derived algebraically for effective inter-arrival and service intervals, for waiting time, for busy period and for output.

Stochastic bounds for the single-server queue

1976 ◽  
Vol 80 (3) ◽  
pp. 521-525 ◽  
Author(s):  
J. Köllerström
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Infinite Divisibility ◽  
Single Server ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Exponential Tail ◽  
Server Queue ◽  
Fitting In

Various elegant properties have been found for the waiting time distribution G for the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds for G which augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).

Limit theorems for the single server queue with traffic intensity one

1970 ◽  
Vol 7 (01) ◽  
pp. 227-233 ◽  
Author(s):  
N. U. Prabhu
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Single Server ◽  
Traffic Intensity ◽  
Arrival Times ◽  
Scale Parameters ◽  
Server Queue ◽  
Queue Discipline ◽  
Image Position ◽  
Queueing Processes

We consider a single server queueing system with inter-arrival times {un , n ≧ 1}, and service times {υn , n ≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un } and {υn } are two independent renewal processes, and 0 &gt; E(un ) =a &lt; ∞, 0 &lt; E(υn ) = b &lt; ∞. The traffic intensity is P ρ b/a(0 &gt; ρ &gt; ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-length Q(t) and waiting time W(t) diverge to + ∞ in distribution as t → ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions for Q(t) and W(t) with appropriate location and scale parameters in the cases P ≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variables Xn = υn – un (n ≧ 1). Let S 0 ≡ 0, Sn = X 1 + X 2 + ··· + Xn (n ≧ 1), and define the sequence {Nk , k ≧ 0} as

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (2) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

Estimation of traffic intensity from queue length data in a deterministic single server queueing system

2021 ◽  
pp. 113693
Author(s):  
Saroja Kumar Singh ◽  
Sarat Kumar Acharya ◽  
Frederico R.B. Cruz ◽  
Roberto C. Quinino
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Single Server ◽  
Traffic Intensity ◽  
Length Data

A single server queue in discrete time with customers served in random order

1972 ◽  
Vol 9 (04) ◽  
pp. 862-867
Author(s):  
D. W. Balmer
Keyword(s):  
Discrete Time ◽  
Random Order ◽  
Single Server ◽  
Queueing Model ◽  
Traffic Intensity ◽  
Single Server Queue ◽  
Server Queue

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ &gt; 0, the condition ρ &lt; ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).

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