On interchangeability for exponential single-server queues in tandem

Consider two exponential single-server queues in tandem and suppose that service rates of customer n are λ n and μ n respectively. In this note, a simple and direct proof is given of the fact that the departure process from the tandem queue is statistically unaffected when the service rates are interchanged if λ n – μn is independent of n. The proof is based only on the memoryless property of exponential distributions.

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Stochastic comparisons for fork-join queues with exponential processing times

Journal of Applied Probability ◽

10.2307/3215387 ◽

1997 ◽

Vol 34 (2) ◽

pp. 487-497 ◽

Cited By ~ 1

Author(s):

Esther Frostig ◽

Tapani Lehtonen

Keyword(s):

Random Variables ◽

Single Server ◽

Departure Process ◽

Stochastic Comparisons ◽

Processing Times ◽

Service Rates ◽

Service Times ◽

Stochastic Majorization

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

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On the ordering of tandem queues with exponential servers

Journal of Applied Probability ◽

10.1017/s0021900200106321 ◽

1986 ◽

Vol 23 (01) ◽

pp. 115-129 ◽

Cited By ~ 4

Author(s):

Tapani Lehtonen

Keyword(s):

Service Time ◽

Queueing System ◽

Arrival Process ◽

Single Server ◽

Tandem Queues ◽

Departure Process ◽

Service Stations ◽

Service Rates

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

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The theory of networks of single server queues and the tandem queue model

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953397000427 ◽

1997 ◽

Vol 10 (4) ◽

pp. 363-381

Author(s):

Pierre Le Gall

Keyword(s):

Traffic Simulation ◽

Heavy Traffic ◽

Telecommunication Networks ◽

Single Server ◽

Tandem Queue ◽

Single Server Queues ◽

Queueing Delay ◽

Jitter Delay ◽

Service Times ◽

Highly Correlated

We consider the stochastic behavior of networks of single server queues when successive service times of a given customer are highly correlated. The study is conducted in two particular cases: 1) networks in heavy traffic, and 2) networks in which all successive service times have the same value (for a given customer), in order to avoid the possibility of breaking up the busy periods. We then show how the local queueing delay (for an arbitrary customer) can be derived through an equivalent tandem queue on the condition that one other local queueing delay is added: the jitter delay due to the independence of partial traffic streams.We consider a practical application of the results by investigating the influence of long packets on the queueing delay of short packets in modern packet switched telecommunication networks. We compare these results with the results given by traffic simulation methods to conclude that there is good agreement between results of calculation and of traffic simulation.

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Stochastic comparisons for fork-join queues with exponential processing times

Journal of Applied Probability ◽

10.1017/s0021900200101111 ◽

1997 ◽

Vol 34 (02) ◽

pp. 487-497 ◽

Cited By ~ 1

Author(s):

Esther Frostig ◽

Tapani Lehtonen

Keyword(s):

Random Variables ◽

Single Server ◽

Departure Process ◽

Stochastic Comparisons ◽

Processing Times ◽

Service Rates ◽

Service Times ◽

Stochastic Majorization

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.

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On the ordering of tandem queues with exponential servers

Journal of Applied Probability ◽

10.2307/3214121 ◽

1986 ◽

Vol 23 (1) ◽

pp. 115-129 ◽

Cited By ~ 12

Author(s):

Tapani Lehtonen

Keyword(s):

Service Time ◽

Queueing System ◽

Arrival Process ◽

Single Server ◽

Tandem Queues ◽

Departure Process ◽

Service Stations ◽

Service Rates

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

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Non-cooperative queueing games on a network of single server queues

Queueing Systems ◽

10.1007/s11134-020-09681-9 ◽

2021 ◽

Author(s):

Corine M. Laan ◽

Judith Timmer ◽

Richard J. Boucherie

Keyword(s):

Nash Equilibrium ◽

Pure Strategy ◽

Single Server ◽

Strategy Space ◽

Single Server Queues ◽

Best Response ◽

Multiple Routes ◽

Service Rates ◽

Expected Sojourn Time ◽

Strategy Nash Equilibrium

AbstractThis paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a $$2\times 2$$ 2 × 2 -grid, and (iv) 2-player games on an $$A\times B$$ A × B -grid with small differences in the service rates.

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A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

Mathematical Problems in Engineering ◽

10.1155/2017/3298605 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Ekaterina Evdokimova ◽

Sabine Wittevrongel ◽

Dieter Fiems

Keyword(s):

Performance Measures ◽

Queueing Systems ◽

Poisson Processes ◽

Series Expansions ◽

Service Rate ◽

Single Server ◽

Queueing Model ◽

State Space Explosion ◽

Finite Queues ◽

Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

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