Asymptotic distribution for the coupon — collector's and sampling-tagging problems, when tagging affects catchability

1975 ◽  
Vol 12 (03) ◽  
pp. 625-628
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Waiting Time ◽  
Limiting Distributions

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN (γ, k) and SN (γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN (γ, k) and SN (γ, j), in terms of the rates at which k and j tend to infinity with N.

Asymptotic distribution for the coupon — collector's and sampling-tagging problems, when tagging affects catchability

1975 ◽  
Vol 12 (3) ◽  
pp. 625-628 ◽  
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Waiting Time ◽  
Limiting Distributions

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN(γ, k) and SN(γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN (γ, k) and SN(γ, j), in terms of the rates at which k and j tend to infinity with N.

The G/M/m queue with finite waiting room

1975 ◽  
Vol 12 (4) ◽  
pp. 779-792 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Asymptotic Distribution ◽  
Waiting Time ◽  
Joint Distribution ◽  
Transient Solution ◽  
Waiting Room ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
The Moment ◽  
Number Of Customers

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

scholarly journals Boundedly rational quasi-Bayesian learning in coordination games with imperfect monitoring

2006 ◽  
Vol 43 (2) ◽  
pp. 335-350 ◽  
Author(s):  
Hsiao-Chi Chen ◽  
Yunshyong Chow ◽  
June Hsieh
Keyword(s):  
Population Size ◽  
Learning Process ◽  
Bayesian Learning ◽  
Coordination Games ◽  
Imperfect Monitoring ◽  
Limiting Distributions ◽  
Long Run ◽  
Time Period ◽  
Extract Information

In this paper we study players' long-run behaviors in evolutionary coordination games with imperfect monitoring. In each time period, signals corresponding to players' underlying actions, instead of the actions themselves, are observed. A boundedly rational quasi-Bayesian learning process is proposed to extract information from the realized signals. We find that players' long-run behaviors depend not only on the correlations between actions and signals, but on the initial probabilities of risk-dominant and non-risk-dominant equilibria being chosen. The conditions under which risk-dominant equilibrium, non-risk-dominant equilibrium, and the coexistence of both equilibria emerges in the long run are shown. In some situations, the number of limiting distributions grows unboundedly as the population size grows to infinity.

On the asymptotic distribution of the size of a stochastic epidemic

1983 ◽  
Vol 20 (2) ◽  
pp. 390-394 ◽  
Author(s):  
Thomas Sellke
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Original Number ◽  
Intuitive Proof

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

The age of a Galton-Watson population with a geometric offspring distribution

2002 ◽  
Vol 39 (4) ◽  
pp. 816-828 ◽  
Author(s):  
F. C. Klebaner ◽  
S. Sagitov
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Initial Population ◽  
Asymptotic Results ◽  
The Past ◽  
Offspring Distribution ◽  
Initial Population Size ◽  
Watson Process ◽  
Looking Back

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

Some extensions of Takács's limit theorems

1974 ◽  
Vol 11 (4) ◽  
pp. 752-761 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Waiting Times ◽  
Interarrival Time ◽  
Positive Probability ◽  
Limiting Distributions ◽  
Waiting Room ◽  
Poisson Arrivals

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

Some extensions of Takács's limit theorems

1974 ◽  
Vol 11 (04) ◽  
pp. 752-761
Author(s):  
D. N. Shanbhag
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Waiting Times ◽  
Interarrival Time ◽  
Positive Probability ◽  
Limiting Distributions ◽  
Waiting Room ◽  
Poisson Arrivals

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

A single-server queue with limited virtual waiting time

1974 ◽  
Vol 11 (03) ◽  
pp. 612-617 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Waiting Time ◽  
Single Server ◽  
Single Server Queue ◽  
Limiting Distributions ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
General Service

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

scholarly journals Boundedly rational quasi-Bayesian learning in coordination games with imperfect monitoring

2006 ◽  
Vol 43 (02) ◽  
pp. 335-350 ◽  
Author(s):  
Hsiao-Chi Chen ◽  
Yunshyong Chow ◽  
June Hsieh
Keyword(s):  
Population Size ◽  
Learning Process ◽  
Bayesian Learning ◽  
Coordination Games ◽  
Imperfect Monitoring ◽  
Limiting Distributions ◽  
Long Run ◽  
Time Period ◽  
Extract Information

In this paper we study players' long-run behaviors in evolutionary coordination games with imperfect monitoring. In each time period, signals corresponding to players' underlying actions, instead of the actions themselves, are observed. A boundedly rational quasi-Bayesian learning process is proposed to extract information from the realized signals. We find that players' long-run behaviors depend not only on the correlations between actions and signals, but on the initial probabilities of risk-dominant and non-risk-dominant equilibria being chosen. The conditions under which risk-dominant equilibrium, non-risk-dominant equilibrium, and the coexistence of both equilibria emerges in the long run are shown. In some situations, the number of limiting distributions grows unboundedly as the population size grows to infinity.

A single-server queue with limited virtual waiting time

1974 ◽  
Vol 11 (3) ◽  
pp. 612-617 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Waiting Time ◽  
Single Server ◽  
Single Server Queue ◽  
Limiting Distributions ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
General Service

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

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