The ballot theorem

Nov/Dec 2016 ◽  
2019 ◽  
Keyword(s):  
Ballot Theorem

Ballots, queues and random graphs

1989 ◽  
Vol 26 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

A derivation of the ballot theorem from the Spitzer–Pollaczek identity

1969 ◽  
Vol 65 (3) ◽  
pp. 755-757 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Walk ◽  
General Theory ◽  
Special Class ◽  
Short Note ◽  
Extensive Literature ◽  
Comprehensive List ◽  
Reduction Problem ◽  
Ballot Theorem ◽  
Ballot Problems

Over a period of many years there has developed an extensive literature on ballot problems. These, in effect, constitute a very special class of random walk problems, and their recent continued development has been justified by the apparent difficulty of reducing expressions given by the general theory down to the very simple ones that it is possible to obtain in an elementary fashion. In this short note we show that the obstacle presented by this reduction problem is actually a rather small one. For background to the above comments, together with a fairly comprehensive list of references to the ballot theory and its attendant applications, the reader is referred to Takács(2).

Ballots, queues and random graphs

1989 ◽  
Vol 26 (01) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γ n (p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 &lt; p &lt; 1. Denote by ρ n (s) the number of vertices in the union of all those components of Γ n (p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n (s) and the limit distribution of ρ n (s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

The ballot theorem strikes again: packet loss process distribution

2000 ◽  
Vol 46 (7) ◽  
pp. 2588-2595 ◽  
Author(s):  
O. Gurewitz ◽  
M. Sidi ◽  
S. Cidon
Keyword(s):  
Packet Loss ◽  
Loss Process ◽  
Ballot Theorem

scholarly journals A proof of Pollaczek-Spitzer identity

1990 ◽  
Vol 13 (4) ◽  
pp. 737-739 ◽  
Author(s):  
S. Paramasamy
Keyword(s):  
Ballot Theorem

In this note we derive a proof of Pollaczek-Spitzer identity using a generalization of Takacs ballot theorem.

Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues

1994 ◽  
Vol 31 (A) ◽  
pp. 251-267
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Random Walks ◽  
Queueing Theory ◽  
Queue Length ◽  
Heuristic Method ◽  
Length Distribution ◽  
Arrival Rate ◽  
Brownian Excursion ◽  
Queue Length Distribution ◽  
Ballot Theorem ◽  
Exit Probabilities

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion. The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

Ballot theorem and lattice path crossings

1978 ◽  
Vol 6 (1) ◽  
pp. 87-90
Author(s):  
Malcolm Kern ◽  
Stanley Walter
Keyword(s):  
Lattice Path ◽  
Ballot Theorem

Direct consequences of the basic Ballot Theorem

2011 ◽  
Vol 81 (10) ◽  
pp. 1476-1481 ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Ballot Theorem
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