scholarly journals Stability of Parallel Server Systems

2021 ◽  
Author(s):  
Pascal Moyal ◽  
Ohad Perry
Keyword(s):  
Queueing System ◽  
Waiting Times ◽  
Theoretical Problem ◽  
Traffic Intensity ◽  
Parallel Server ◽  
Routing Policy ◽  
Server Systems ◽  
Routing Policies ◽  
The Stability ◽  
Server System

A parallel server system is a queueing system in which jobs are routed upon their arrivals to one of several buffers, each handled by a different server. The main operational and theoretical problem in such systems is to find an efficient routing policy that maximizes their throughput (or minimizes waiting times). The paper “Stability of Parallel Server Systems” considers a large class of routing policies, which includes the most prevalent policies studies in the literature, under the assumption that routing errors may occur because ofincomplete information about the state of the system at decision epochs. The stability region for this class of policies is studied as a function of the error probability, and it is shown that the standard stability condition, namely, that the traffic intensity is smaller than one, does not guarantee that the system is stable.

The virtual waiting time of the GI/G/1 queue in heavy traffic

1971 ◽  
Vol 3 (2) ◽  
pp. 249-268 ◽  
Author(s):  
E. Kyprianou
Keyword(s):  
Markov Chains ◽  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Double Sequence ◽  
Traffic Intensity ◽  
Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

The virtual waiting time of the GI/G/1 queue in heavy traffic

1971 ◽  
Vol 3 (02) ◽  
pp. 249-268 ◽  
Author(s):  
E. Kyprianou
Keyword(s):  
Markov Chains ◽  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Double Sequence ◽  
Traffic Intensity ◽  
Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {W n i } of a sequence {Q i } of stable GI/G/1 queues, where W n i is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρ i ↑ 1 as i → ∞. Here &rH i is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {W n i } and obtained limit theorems in the three cases when n 1/2(ρ i -1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {W n i } and obtained limits in the two cases when n 1/2(ρ i − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

scholarly journals STABILITY ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

2014 ◽  
Vol 31 (02) ◽  
pp. 1440002 ◽  
Author(s):  
K. AVRACHENKOV ◽  
E. MOROZOV ◽  
R. NEKRASOVA ◽  
B. STEYAERT
Keyword(s):  
Queueing System ◽  
Stability Domain ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Single Orbit ◽  
Transmission Control Protocols ◽  
Regenerative Approach ◽  
The Stability ◽  
Multi Access

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

Optimal Thresholds of an Infinite Buffer Discrete-Time Two-Server System with Triadic Policy

2011 ◽  
Vol 2 (4) ◽  
pp. 75-88
Author(s):  
Veena Goswami ◽  
G. B. Mund
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Minimum Cost ◽  
Service Rate ◽  
Closed Form Solutions ◽  
Optimal Thresholds ◽  
Optimal Service ◽  
Number Of Customers ◽  
System Operating ◽  
Server System

This paper analyzes a discrete-time infinite-buffer Geo/Geo/2 queue, in which the number of servers can be adjusted depending on the number of customers in the system one at a time at arrival or at service completion epoch. Analytical closed-form solutions of the infinite-buffer Geo/Geo/2 queueing system operating under the triadic (0, Q N, M) policy are derived. The total expected cost function is developed to obtain the optimal operating (0, Q N, M) policy and the optimal service rate at minimum cost using direct search method. Some performance measures and sensitivity analysis have been presented.

A perpetuity and the M/M/∞ ranked server system

1997 ◽  
Vol 34 (02) ◽  
pp. 508-513 ◽  
Author(s):  
J. Preater
Keyword(s):  
Difference Equation ◽  
Generating Function ◽  
Classical Result ◽  
Equilibrium Size ◽  
Server Systems ◽  
Random Difference Equation ◽  
Random Difference ◽  
Server System

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.

On the comparison of waiting times in tandem queues

1981 ◽  
Vol 18 (03) ◽  
pp. 707-714 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Waiting Times ◽  
Distribution Functions ◽  
Partial Ordering ◽  
Tandem Queues ◽  
Order Relations ◽  
Service Distribution ◽  
In Series ◽  
Definition Of

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

scholarly journals The Multiphase Queuing System of the Rijeka Airport

Pomorstvo ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 205-209
Author(s):  
Svjetlana Hess ◽  
Ana Grbčić
Keyword(s):  
Performance Indicators ◽  
Queueing Theory ◽  
Rare Case ◽  
Queueing System ◽  
Service System ◽  
Waiting Times ◽  
Real Problem ◽  
Single Server ◽  
Scientific Contribution ◽  
The Real

The paper gives an overview of the real system as a multiphase single server queuing problem, which is a rare case in papers dealing with the application of the queueing theory. The methodological and scientific contribution of this paper is primarily in setting up the model of the real problem applying the multiphase queueing theory. The research of service system at Rijeka Airport may allow the airport to be more competitive by increasing service quality. The existing performance measures have been evaluated in order to improve Rijeka Airport queueing system, as a record number of passengers is to be expected in the next few years. Performance indicators have pointed out how the system handles congestion. The research is also focused on defining potential bottlenecks and comparing the results with IATA guidelines in terms of maximum waiting times.

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