Combinatorics of geometrically distributed random variables: Left-to-right maxima

Prodinger, Helmut

doi:10.1016/0012-365x(95)00141-i

Cited by 56 publications

(45 citation statements)

References 8 publications

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“…Therefore, results for m-records in permutations as obtained, e.g., by Prodinger [12] are of interest here also, and vice versa, our detailed studies of this hiring strategy might lead to new insights in connection with record statistics.…”

Section: Introductionmentioning

confidence: 80%

“…Recall the example mentioned in the introduction: for m = 3 and σ 7 = 4617352, the hiring set contains {4, 6, 1, 7, 5} hence h 7,3 = 5. Then, for 1 ≤ m ≤ n, Archibald and Martínez [1] showed (see also [12] for corresponding results on m-records) the following exact result for the expectation of h n,m , where the given asymptotic expansion holds uniformly for 1 ≤ m ≤ n:…”

Section: Size Of the Hiring Setmentioning

confidence: 99%

“…Thus, the probability generating function h n,m (v) := ≥0 P {h n,m = } v is given by the following explicit formula (note that the corresponding probability generating function for m-records in permutations already appears in [12]), which will be the starting point to derive the exact and asymptotic results: …”

Section: Proof Of Theoremmentioning

confidence: 99%

“…, x n of n different scores, which are ranked x i1 < x i2 < · · · < x in . In the usual definition of m-records (see [12]) and references therein), an element x i is contained in the set R m of m-records if there exists an index j ≥ i, such that x i is the m-th largest element in the set {x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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Hiring above the m-th Best Candidate: A Generalization of Records in Permutations

Helmi

Martínez

Panholzer

2012

LATIN 2012: Theoretical Informatics

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Abstract. The hiring problem is a simple model of on-line decisionmaking under uncertainty. As in many other such models, the input is a sequence of instances and a decision must be taken for each instance depending on the subsequence examined so far, while nothing is known about the future. One famous example of on-line decision-making the secretary problem, formally introduced in the early sixties. Broder et al. (2008) introduced the hiring problem as an extension of the secretary problem. Instead of selecting only one candidate, we are looking for selecting (hiring) many candidates to grow up a small company. In this context, a hiring strategy should meet two demands: to hire candidates at some reasonable rate and to improve the average quality of the hired staff. Soon afterwards, Archibald and Martínez (2009) introduced a discrete model of the hiring problem where candidates seen so far could be ranked from best to worst without the need to know their absolute quality scores. Hence the sequence of candidates could be modeled as a random permutation. Two general families of hiring strategies were introduced: hiring above the m-th best candidate and hiring in the top P % quantile (for instance, P = 50 is hiring above the median). In this paper we consider only hiring above the m-th best candidate. We introduce new hiring parameters that describe the dynamics of the hiring process, like the distance between the last two hirings, and the quality of the hired staff, like the score of the best discarded candidate. While Archibald and Martínez made systematic use of analytic combinatorics techniques (Flajolet, Sedgewick, 2008) in their analysis, we use here a different approach to study the various hiring parameters related associated to the hiring process. We are able to obtain explicit formulas for the probability distribution or the probability generating function of the random variables of interest in a rather direct way. The explicit nature of our results also allows a very detailed study of their asymptotic behaviour. Adding our new results to those of Archibald and Martínez leads to a very precise This work started when the first author was visiting the third author in a short stay supported by an FPI grant from the Spanish Ministry of Science. The first and the second authors were supported by project TIN2010-17254 (FRADA) from the Spanish Ministry of Science and Innovation. The third author was supported by the Austrian Science Foundation FWF, grant S9608-N23. quantitative characterization of the hiring above the m-th best candidate strategy. This might prove very useful in applications of the hiring process, e.g., in data stream algorithms.

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Section: Introductionmentioning

confidence: 80%

Section: Size Of the Hiring Setmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hiring above the m-th Best Candidate: A Generalization of Records in Permutations

Helmi

Martínez

Panholzer

2012

LATIN 2012: Theoretical Informatics

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“…Thus, the expected time interval between the issuance of the first NAK and reception of last bit of last retransmission spurt can be obtained as (5) Note that is the expected total time taken for transmission of meta and file-data PDUs until all of them have been successfully delivered, minus the time taken to transmit them for their first trials. Thus (6) Note that in deferred NAK mode, the receiver sends NAK only after receiving EOF PDU. Therefore, expected file-delivery time of a transaction, which includes the expected EOF delivery time, is given as (7) To complete the analysis, we need to obtain .…”

Section: Expected File-delivery Timementioning

confidence: 99%

Expected File-Delivery Time of Deferred NAK ARQ in CCSDS File-Delivery Protocol

Lee

Baek

2004

IEEE Trans. Commun.

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Abstract-We analyze an automatic repeat-request (ARQ) scheme of the Consultative Committee for Space Data Systems file-delivery protocol for the single-hop file-transfer operation. With regard to performance measures, this paper is mainly concerned with the time taken to transfer a file (file-delivery time) and throughput efficiency. We discuss the ARQ timer-setting rule that minimizes the expected file-delivery time under the constraint that the throughput efficiency is maximized. Then, for that timer-setting rule, we derive the expected file-delivery time.

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Normal approximations of the number of records in geometrically distributed random variables

Bai

Hwang

Wei

1998

Random Struct. Alg.

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Combinatorics of geometrically distributed random variables: Left-to-right maxima

Cited by 56 publications

References 8 publications

Hiring above the m-th Best Candidate: A Generalization of Records in Permutations

Hiring above the m-th Best Candidate: A Generalization of Records in Permutations

Expected File-Delivery Time of Deferred NAK ARQ in CCSDS File-Delivery Protocol

Normal approximations of the number of records in geometrically distributed random variables

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