The theory of periodic screening I: Lead time and proportion detected

1976 ◽  
Vol 8 (1) ◽  
pp. 127-143 ◽  
Author(s):  
Philip C. Prorok
Keyword(s):  
Chronic Disease ◽  
Natural History ◽  
Time Distribution ◽  
Lead Time ◽  
Screening Program ◽  
Clinical State ◽  
Recurrence Time ◽  
Sojourn Time Distribution ◽  
History Of ◽  
Forward Recurrence Time

A stochastic model for a periodic screening program is presented in which the natural history of a chronic disease is assumed to follow a progressive path from a preclinical state to a clinical state. The sampling of preclinical state sojourn times by screening examinations generates bounds on the preclinical state recurrence times. The distribution of the bounded forward recurrence time is derived and used to obtain the distribution and mean of the lead time, and relationships for calculating the proportion of preclinical cases detected. These expressions are derived in terms of the preclinical state sojourn-time distribution and adjustible parameters important in the design of a periodic screening program.

A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times

1985 ◽  
Vol 22 (02) ◽  
pp. 419-428 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Sample Path ◽  
The Other ◽  
Independent Random Variables ◽  
Recurrence Time ◽  
Stochastic Decomposition ◽  
Set Up ◽  
Forward Recurrence Time ◽  
Main Model

In this note we prove some stochastic decomposition results for variations of theGI/G/1 queue. Our main model is aGI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on. Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of Gelenbe and Iasnogorodski [5]. We use sample path arguments, which are also used to prove stochastic decomposition in aGI/G/1 queue with set-up time.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (1) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Point processes generated by transitions of Markov chains

1973 ◽  
Vol 5 (2) ◽  
pp. 262-286 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Point Processes ◽  
Queueing System ◽  
Initial Distribution ◽  
Recurrence Time ◽  
Asynchronous Sampling ◽  
Doubly Stochastic ◽  
Special Cases ◽  
Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

Some non-stationary point processes with stationary forward recurrence time distribution

1975 ◽  
Vol 12 (1) ◽  
pp. 167-169 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Stationary Point ◽  
Time Distribution ◽  
Point Processes ◽  
Recurrence Time ◽  
Renewal Processes ◽  
Forward Recurrence Time ◽  
Stationary Point Processes

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.

Further monotonicity properties of renewal processes

1992 ◽  
Vol 24 (03) ◽  
pp. 575-588 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Continuous Time ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Recurrence Time ◽  
Renewal Time ◽  
Hazard Rate Ordering ◽  
Forward Recurrence Time ◽  
Decreasing Failure Rate ◽  
Monotonicity Results

In a discrete-time renewal process {Nk, k= 0, 1, ·· ·}, letZkandAkbe the forward recurrence time and the renewal age, respectively, at timek.In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k= 0, 1, ·· ·} and {Zk, k= 0, 1, ·· ·} are monotonically non-decreasing inkin hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m– Nk, k= 0, 1, ·· ·} to be non-increasing inkin hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

Point processes generated by transitions of Markov chains

1973 ◽  
Vol 5 (02) ◽  
pp. 262-286 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Point Processes ◽  
Queueing System ◽  
Initial Distribution ◽  
Recurrence Time ◽  
Asynchronous Sampling ◽  
Doubly Stochastic ◽  
Special Cases ◽  
Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

The theory of periodic screening I: Lead time and proportion detected

1976 ◽  
Vol 8 (01) ◽  
pp. 127-143 ◽  
Author(s):  
Philip C. Prorok
Keyword(s):  
Chronic Disease ◽  
Natural History ◽  
Time Distribution ◽  
Lead Time ◽  
Screening Program ◽  
Clinical State ◽  
Recurrence Time ◽  
Sojourn Time Distribution ◽  
History Of ◽  
Forward Recurrence Time

A stochastic model for a periodic screening program is presented in which the natural history of a chronic disease is assumed to follow a progressive path from a preclinical state to a clinical state. The sampling of preclinical state sojourn times by screening examinations generates bounds on the preclinical state recurrence times. The distribution of the bounded forward recurrence time is derived and used to obtain the distribution and mean of the lead time, and relationships for calculating the proportion of preclinical cases detected. These expressions are derived in terms of the preclinical state sojourn-time distribution and adjustible parameters important in the design of a periodic screening program.

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