Inequalities Via Majorization and Weak Majorization

1980 ◽  
pp. 102-141
Author(s):  
Y.L. TONG
Keyword(s):  
Weak Majorization

scholarly journals Multiple-server system with flexible arrivals

2011 ◽  
Vol 43 (4) ◽  
pp. 985-1004 ◽  
Author(s):  
Osman T. Akgun ◽  
Rhonda Righter ◽  
Ronald Wolff
Keyword(s):  
Finite Buffers ◽  
Service Production ◽  
Weak Majorization ◽  
Service Rates ◽  
Join The Shortest Queue ◽  
Traffic Systems ◽  
Shortest Queue ◽  
Performance Gains ◽  
Number Of Customers ◽  
Server System

In many service, production, and traffic systems there are multiple types of customers requiring different types of ‘servers’, i.e. different services, products, or routes. Often, however, a proportion of the customers are flexible, i.e. they are willing to change their type in order to achieve faster service, and even if this proportion is small, it has the potential of achieving large performance gains. We generalize earlier results on the optimality of ‘join the shortest queue’ (JSQ) for flexible arrivals to the following: arbitrary arrivals where only a subset are flexible, multiple-server stations, and abandonments. Surprisingly, with abandonments, the optimality of JSQ for minimizing the number of customers in the system depends on the relative abandonment and service rates. We extend our model to finite buffers and resequencing. We assume exponential service. Our optimality results are very strong; we minimize the queue length process in the weak majorization sense.

Sample Path Criteria for Weak Majorization

1994 ◽  
Vol 26 (01) ◽  
pp. 155-171 ◽  
Author(s):  
Panayotis D. Sparaggis ◽  
Don Towsley ◽  
Christos G. Cassandras
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Sample Path ◽  
Cumulative Number ◽  
Stochastic Orderings ◽  
Optimal Policies ◽  
Weak Majorization ◽  
Queue Lengths ◽  
And Performance

We present two forms of weak majorization, namely, very weak majorization and p-weak majorization that can be used as sample path criteria in the analysis of queueing systems. We demonstrate how these two criteria can be used in making comparisons among the joint queue lengths of queueing systems with blocking and/or multiple classes, by capturing an interesting interaction between state and performance descriptors. As a result, stochastic orderings on performance measures such as the cumulative number of losses can be derived. We describe applications that involve the determination of optimal policies in the context of load-balancing and scheduling.

scholarly journals Weak majorization inequalities and convex functions

2003 ◽  
Vol 369 ◽  
pp. 217-233 ◽  
Author(s):  
Jaspal Singh Aujla ◽  
Fernando C. Silva
Keyword(s):  
Convex Functions ◽  
Weak Majorization

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.

A relation of weak majorization and its applications to certain inequalities for means

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
Shan-He Wu ◽  
Huan-Nan Shi
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Weak Majorization

AbstractA relation of weak majorization for n-dimensional real vectors is established, the result is then used to derive some inequalities involving the power mean, the arithmetic mean and the geometric mean in n variables.

Optimum continuous block designs

1981 ◽  
Vol 377 (1771) ◽  
pp. 405-416 ◽  
Keyword(s):  
Incidence Matrix ◽  
Block Design ◽  
Information Matrix ◽  
Block Designs ◽  
Negative Numbers ◽  
Weak Majorization ◽  
Continuous Block ◽  
Complete Specification

The incidence matrix of a block design is replaced by a normalized version, N, in which the entries are non-negative numbers whose sum is unity. The so-called C-matrix, the information matrix for estimation of treatm ent contrasts, is similarly replaced by the normalized analogue C(N). We study the set of ordered eigenvalues of all C(N) and give a complete specification for three treatments (rows). For any number of treatm ents we characterize the eigenvalues of an im portant subclass of designs for which the non-zero entries in any given block are equal. It is suggested that the natural ordering between designs is upper weak majorization of the eigenvalues. Using this we show how to improve a given N-matrix and this leads to several optimality statements.

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