Braess's paradox in a queueing network with state-dependent routing

1997 ◽  
Vol 34 (1) ◽  
pp. 134-154 ◽  
Author(s):  
Bruce Calvert ◽  
Wiremu Solomon ◽  
Ilze Ziedins
Keyword(s):  
Queueing Network ◽  
Wheatstone Bridge ◽  
The Other ◽  
Time Of Arrival ◽  
Expected Time ◽  
Initial Network ◽  
Braess’S Paradox ◽  
State Dependent ◽  
Service Rates ◽  
Queue Lengths

We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential.We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them.We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.

Braess's paradox in a queueing network with state-dependent routing

1997 ◽  
Vol 34 (01) ◽  
pp. 134-154 ◽  
Author(s):  
Bruce Calvert ◽  
Wiremu Solomon ◽  
Ilze Ziedins
Keyword(s):  
Queueing Network ◽  
Wheatstone Bridge ◽  
The Other ◽  
Time Of Arrival ◽  
Expected Time ◽  
Initial Network ◽  
Braess’S Paradox ◽  
State Dependent ◽  
Service Rates ◽  
Queue Lengths

We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential. We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them. We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.

Throughput properties of a queueing network with distributed dynamic routing and flow control

1996 ◽  
Vol 28 (01) ◽  
pp. 285-307 ◽  
Author(s):  
Leandros Tassiulas ◽  
Anthony Ephremides
Keyword(s):  
Flow Control ◽  
Rate Control ◽  
Queueing Network ◽  
Local State ◽  
Service Rate ◽  
Maximum Throughput ◽  
State Information ◽  
Arbitrary Topology ◽  
State Dependent ◽  
Service Rates

A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.

The effect of increasing service rates in a closed queueing network

1986 ◽  
Vol 23 (2) ◽  
pp. 474-483 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Likelihood Ratio ◽  
Queueing Network ◽  
Stochastic Ordering ◽  
Product Form ◽  
Closed Queueing Network ◽  
Equilibrium Behavior ◽  
Service Rates ◽  
Queue Lengths ◽  
Multivariate Likelihood

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

Extremal properties of the shortest/longest non-full queue policies in finite-capacity systems with state-dependent service rates

1993 ◽  
Vol 30 (01) ◽  
pp. 223-236 ◽  
Author(s):  
P. D. Sparaggis ◽  
D. Towsley ◽  
C. G. Cassandras
Keyword(s):  
Optimal Allocation ◽  
Buffer Allocation ◽  
Performance Indices ◽  
Parallel Queues ◽  
State Dependent ◽  
Weak Majorization ◽  
Service Rates ◽  
Queue Lengths ◽  
Buffer Capacities ◽  
Extremal Properties

We consider the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities. Service rates are state-dependent and non-decreasing with respect to queue lengths. We establish the extremal properties of the shortest non-full queue (SNQ) and the longest non-full queue (LNQ) policies, in systems with concave/convex service rates. Our analysis is based on the weak majorization of joint queue lengths which leads to stochastic orderings of critical performance indices. Moreover, we solve the buffer allocation problem, i.e. the problem of how to distribute a number of buffers among the queues. The two optimal allocation schemes are also ‘extreme', in the sense of capacity balancing. Some extensions are also discussed.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (1) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

The effect of increasing service rates in a closed queueing network

1986 ◽  
Vol 23 (02) ◽  
pp. 474-483 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Likelihood Ratio ◽  
Queueing Network ◽  
Stochastic Ordering ◽  
Product Form ◽  
Closed Queueing Network ◽  
Equilibrium Behavior ◽  
Service Rates ◽  
Queue Lengths ◽  
Multivariate Likelihood

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

On Generalized Networks of Queues with Positive and Negative Arrivals

1993 ◽  
Vol 7 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Xiuli Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Network ◽  
Form Solution ◽  
Product Form ◽  
Current State ◽  
State Dependent ◽  
Regular Customer ◽  
Generalized Networks ◽  
Service Rates ◽  
The Individual ◽  
Departure Processes

Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe [5] showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.

Throughput properties of a queueing network with distributed dynamic routing and flow control

1996 ◽  
Vol 28 (1) ◽  
pp. 285-307 ◽  
Author(s):  
Leandros Tassiulas ◽  
Anthony Ephremides
Keyword(s):  
Flow Control ◽  
Rate Control ◽  
Queueing Network ◽  
Local State ◽  
Service Rate ◽  
Maximum Throughput ◽  
State Information ◽  
Arbitrary Topology ◽  
State Dependent ◽  
Service Rates

A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.

Realization factors and sensitivity analysis of queueing networks with state-dependent service rates

1990 ◽  
Vol 22 (01) ◽  
pp. 178-210 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Service Rate ◽  
Steady State Probability ◽  
State Dependent ◽  
Service Rates ◽  
Realization Factors

The paper studies the sensitivity of the throughput with respect to a mean service rate in a closed queueing network with exponentially distributed service requirements and state-dependent service rates. The study is based on perturbation analysis of queueing networks. A new concept, the realization factor of a perturbation, is introduced. The properties of realization factors are discussed, and a set of equations specifying the realization factors are derived. The elasticity of the steady state throughput with respect to a mean service rate equals the product of the steady state probability and the corresponding realization factor. This elasticity can be estimated by applying a perturbation analysis algorithm to a sample path of the system. The sample path elasticity of the throughput with respect to a mean service rate converges with probability 1 to the elasticity of the steady state throughput. The theory provides an analytical method of calculating the throughput sensitivity and justifies the application of perturbation analysis.

Extremal properties of the shortest/longest non-full queue policies in finite-capacity systems with state-dependent service rates

1993 ◽  
Vol 30 (1) ◽  
pp. 223-236 ◽  
Author(s):  
P. D. Sparaggis ◽  
D. Towsley ◽  
C. G. Cassandras
Keyword(s):  
Optimal Allocation ◽  
Buffer Allocation ◽  
Performance Indices ◽  
Parallel Queues ◽  
State Dependent ◽  
Weak Majorization ◽  
Service Rates ◽  
Queue Lengths ◽  
Buffer Capacities ◽  
Extremal Properties

We consider the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities. Service rates are state-dependent and non-decreasing with respect to queue lengths. We establish the extremal properties of the shortest non-full queue (SNQ) and the longest non-full queue (LNQ) policies, in systems with concave/convex service rates. Our analysis is based on the weak majorization of joint queue lengths which leads to stochastic orderings of critical performance indices. Moreover, we solve the buffer allocation problem, i.e. the problem of how to distribute a number of buffers among the queues. The two optimal allocation schemes are also ‘extreme', in the sense of capacity balancing. Some extensions are also discussed.

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