A generalization of the inspection paradox in an ordinary renewal process

Gakis, Konstantinos; Sivazlian, B. D.

doi:10.1080/07362999308809300

Cited by 8 publications

(6 citation statements)

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“…, g κ are monotone non-decreasing and left-continuous. If P(X 1 = α) > 0, then G(x) = c n necessarily has to hold for all nα ≤ x ≤ nω, n ∈ N. Thus, for G as in (7) to be left-continuous at the points α, . .…”

Section: General Resultsmentioning

confidence: 99%

“…Corollary 2 shows that equality for an appropriate value of z is enough to determine the general form of the distribution function G (on finite intervals). Thus, if we have equality of the survival functions of X N(T)+1 and X 1 for this z ∈ S, then T is of the form (7) and Corollary 1 yields X N(T)+1 = st X 1 .…”

Section: Equality Of the Survival Functionsmentioning

confidence: 98%

“…The renewal interval covering t is referred to as the "inspection interval" and its length is given by X N(t)+1 = S N(t)+1 − S N(t) (see Figure 2). Representations for the survival function and the expected value of the inspection interval length X N(t)+1 can be found in the literature (see, e.g., Gakis and Sivazlian [7], pp. 44-45).…”

Section: The Classical Inspection Paradox Inequalitymentioning

confidence: 99%

“…Stats 2024, 7 Much attention has been paid to the study of the inspection paradox, its properties, and implications (see, e.g., Gakis and Sivazlian [7], Angus [8], Ross [9]). In particular, considering a random variable for the time of inspection instead of a deterministic time leads to insights regarding the quantification of the effect (see, e.g., Kamps [10]).…”

Section: Introductionmentioning

confidence: 99%

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On Non-Occurrence of the Inspection Paradox

Rauwolf,

Kamps

2024

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The well-known inspection paradox or waiting time paradox states that, in a renewal process, the inspection interval is stochastically larger than a common interarrival time having a distribution function F, where the inspection interval is given by the particular interarrival time containing the specified time point of process inspection. The inspection paradox may also be expressed in terms of expectations, where the order is strict, in general. A renewal process can be utilized to describe the arrivals of vehicles, customers, or claims, for example. As the inspection time may also be considered a random variable T with a left-continuous distribution function G independent of the renewal process, the question arises as to whether the inspection paradox inevitably occurs in this general situation, apart from in some marginal cases with respect to F and G. For a random inspection time T, it is seen that non-trivial choices lead to non-occurrence of the paradox. In this paper, a complete characterization of the non-occurrence of the inspection paradox is given with respect to G. Several examples and related assertions are shown, including the deterministic time situation.

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Section: General Resultsmentioning

confidence: 99%

Section: Equality Of the Survival Functionsmentioning

confidence: 98%

Section: The Classical Inspection Paradox Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Non-Occurrence of the Inspection Paradox

Rauwolf,

Kamps

2024

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“…The analysis of a residual service time is well-known in the literature, and in the simplest cases is associated with the inspection paradox or waiting time paradox (see e.g. [17,20,21,24,28,31,33] and many others).…”

Section: Remark 25mentioning

confidence: 99%

Continuity theorems for the M/M/1/n queueing system

Abramov

2008

Queueing Syst

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In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.Keywords Continuity theorems · Loss systems · M/GI/1/n and M/M/1/n queues · Busy period · Branching process · Number of level crossings · Kolmogorov (uniform) metric · Stochastic ordering · Stochastic inequalities Mathematics Subject Classification (2000) 60K25 · 60B05 · 60E15 · 62E17 Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.

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Inspection Paradox (Update)

Kamps¹

2004

Encyclopedia of Statistical Sciences

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A generalization of the inspection paradox in an ordinary renewal process

Cited by 8 publications

References 1 publication

On Non-Occurrence of the Inspection Paradox

On Non-Occurrence of the Inspection Paradox

Continuity theorems for the M/M/1/n queueing system

Inspection Paradox (Update)

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