How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (3) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (03) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

scholarly journals Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1893
Author(s):  
Bara Kim ◽  
Jeongsim Kim ◽  
Jerim Kim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Probability Generating Function ◽  
Finite Collection ◽  
Numerical Inversion ◽  
Analytic Method ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
The Matrix ◽  
The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (2) ◽  
pp. 461-466 ◽  
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Estimation of the mean waiting time of a customer subject to balking: a simulation study

2007 ◽  
Vol 19 (1) ◽  
pp. 63-63
Author(s):  
Jaejin Jang ◽  
Jaewoo Chung ◽  
Jungdae Suh ◽  
Jongtae Rhee
Keyword(s):  
Waiting Time ◽  
Simulation Study ◽  
Mean Waiting Time ◽  
The Mean

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (2) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

scholarly journals Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes

2008 ◽  
Vol 45 (02) ◽  
pp. 472-480
Author(s):  
Daniel Tokarev
Keyword(s):  
Generating Function ◽  
Regular Variation ◽  
Probability Generating Function ◽  
Integral Transforms ◽  
Partial Sums ◽  
Finite Variance ◽  
Time To Extinction ◽  
Mean Time To Extinction ◽  
The Mean ◽  
Mean Time

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (02) ◽  
pp. 478-490
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-lcustomer who, upon his arrival, meetskcustomers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-lcustomer who, upon his arrival, meetsk+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

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