Exact results for a secretary problem

1996 ◽  
Vol 33 (03) ◽  
pp. 630-639 ◽  
Author(s):  
M. P. Quine ◽  
J. S. Law
Keyword(s):  
Markov Chain ◽  
Optimal Strategy ◽  
Secretary Problem ◽  
Exact Results ◽  
Asymptotic Results ◽  
Backwards Induction ◽  
Imbedded Markov Chain ◽  
Probability Of Success

We consider the following secretary problem: items ranked from 1 to n are randomly selected without replacement, one at a time, and to ‘win' is to stop at an item whose overall rank is less than or equal to s, given only the relative ranks of the items drawn so far. Our method of analysis is based on the existence of an imbedded Markov chain and uses the technique of backwards induction. In principal the approach can be used to give exact results for any value of s; we do the working for s = 3. We give exact results for the optimal strategy, the probability of success and the distribution of T, and the total number of draws when the optimal strategy is implemented. We also give some asymptotic results for these quantities as n → ∞.

Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain

1966 ◽  
Vol 3 (2) ◽  
pp. 512-520 ◽  
Author(s):  
J. H. Jenkins
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Probability Generating Function ◽  
Joint Distributions ◽  
Left Behind ◽  
Markov Chain Theory ◽  
Imbedded Markov Chain ◽  
Chain Theory

SummaryProbability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

The recurrence and transience of two-dimensional linear birth and death processes

1980 ◽  
Vol 12 (3) ◽  
pp. 615-639 ◽  
Author(s):  
J. Hutton
Keyword(s):  
Markov Chain ◽  
Linear Functions ◽  
Death Process ◽  
Two Dimensional ◽  
Continuous Time Markov Chain ◽  
Birth And Death Process ◽  
Positive Probability ◽  
Birth And Death Processes ◽  
Imbedded Markov Chain ◽  
Transient Case

A two-dimensional linear birth and death process is a continuous-time Markov chain Y(·) with state space (Z+)2 which can jump from the point (n, m) to one of its four neighbors, with rates that are linear functions of n and m. Criteria are extended for determining whether such a process has a positive probability or zero probability of escaping to infinity. In the transient case considered, the projections of the imbedded Markov chain {Xn} of the successive states visited by Y(·) on a suitable pair of orthonormal vectors v and w are shown to be regularly varying sequences with index 1. Specifically, (Xn, v)∽δn and (Xn, w)∽ kn/log n for positive constants δ and k.

COUNTING OF OLIGOMERS IN SEQUENCES GENERATED BY MARKOV CHAINS FOR DNA MOTIF DISCOVERY

2009 ◽  
Vol 07 (01) ◽  
pp. 39-54 ◽  
Author(s):  
GAO SHAN ◽  
WEI-MOU ZHENG
Keyword(s):  
Markov Chain ◽  
Motif Discovery ◽  
Source Code ◽  
The Other ◽  
Promoter Sequences ◽  
Second Moments ◽  
Dna Motif ◽  
Imbedded Markov Chain ◽  
Z Scores ◽  
Dna Motif Discovery

By means of the technique of the imbedded Markov chain, an efficient algorithm is proposed to exactly calculate first, second moments of word counts and the probability for a word to occur at least once in random texts generated by a Markov chain. A generating function is introduced directly from the imbedded Markov chain to derive asymptotic approximations for the problem. Two Z-scores, one based on the number of sequences with hits and the other on the total number of word hits in a set of sequences, are examined for discovery of motifs on a set of promoter sequences extracted from A. thaliana genome. Source code is available at .

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