A note on the cost of carrier-borne, right-shift, epidemic models

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.

Download Full-text

Scheduling jobs on non-identical IFR processors to minimize general cost functions

Advances in Applied Probability ◽

10.1017/s0001867800024010 ◽

1991 ◽

Vol 23 (04) ◽

pp. 909-924 ◽

Cited By ~ 3

Author(s):

Rhonda Righter ◽

Susan H. Xu

Keyword(s):

Cost Function ◽

Optimal Policy ◽

Expected Number ◽

Expected Cost ◽

Processing Times ◽

Special Cases ◽

The Cost ◽

General Cost Functions ◽

Tardy Jobs ◽

Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

Download Full-text

Scheduling jobs on non-identical IFR processors to minimize general cost functions

Advances in Applied Probability ◽

10.2307/1427683 ◽

1991 ◽

Vol 23 (4) ◽

pp. 909-924 ◽

Cited By ~ 10

Author(s):

Rhonda Righter ◽

Susan H. Xu

Keyword(s):

Cost Function ◽

Optimal Policy ◽

Expected Number ◽

Expected Cost ◽

Processing Times ◽

Special Cases ◽

The Cost ◽

General Cost Functions ◽

Tardy Jobs ◽

Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

Download Full-text

On the general bilinear time series model

Journal of Applied Probability ◽

10.2307/3213984 ◽

1988 ◽

Vol 25 (3) ◽

pp. 553-564 ◽

Cited By ~ 42

Author(s):

Jian Liu ◽

Peter J. Brockwell

Keyword(s):

Time Series ◽

Stationary Solution ◽

Time Series Model ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Special Cases ◽

Non Linear ◽

Necessary And Sufficient ◽

Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Download Full-text

On the general bilinear time series model

Journal of Applied Probability ◽

10.1017/s0021900200041279 ◽

1988 ◽

Vol 25 (03) ◽

pp. 553-564 ◽

Cited By ~ 9

Author(s):

Jian Liu ◽

Peter J. Brockwell

Keyword(s):

Time Series ◽

Stationary Solution ◽

Time Series Model ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Special Cases ◽

Non Linear ◽

Necessary And Sufficient ◽

Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Download Full-text

The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay

Dependence Modeling ◽

10.1515/demo-2020-0012 ◽

2020 ◽

Vol 8 (1) ◽

pp. 210-220

Author(s):

Jan-Frederik Mai

Keyword(s):

Conditional Independence ◽

A Priori ◽

Load Sharing ◽

Reliability Theory ◽

Sufficient Condition ◽

Special Cases ◽

De Finetti ◽

Stable Mixture ◽

Special Case ◽

Lévy Subordinators

AbstractWe derive a sufficient condition on the symmetric norm ||·|| such that the probability distribution associated with the density function f (x) ∝exp(−λ ||x||) is conditionally independent and identically distributed in the sense of de Finetti’s seminal theorem. The criterion is mild enough to comprise the ℓp-norms as special cases, in which f is shown to correspond to a polynomially tilted stable mixture of products of transformed Gamma densities. In another special case of interest f equals the density of a time-homogeneous load sharing model, popular in reliability theory, whose motivation is a priori unrelated to the concept of conditional independence. The de Finetti structure reveals a surprising link between time-homogeneous load sharing models and the concept of Lévy subordinators.

Download Full-text

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

Fundamenta Informaticae ◽

10.3233/fi-1981-4305 ◽

1981 ◽

Vol 4 (3) ◽

pp. 531-549 ◽

Cited By ~ 1

Author(s):

Miklós Szijártó

Keyword(s):

Formal Languages ◽

Parallel Program ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Concurrent System ◽

Closure Properties ◽

Sequential Program ◽

Necessary And Sufficient ◽

Special Case ◽

Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

Download Full-text

Some analyses on the control of queues using level crossings of regenerative processes

Journal of Applied Probability ◽

10.2307/3212974 ◽

1980 ◽

Vol 17 (3) ◽

pp. 814-821 ◽

Cited By ~ 20

Author(s):

J. G. Shanthikumar

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Queueing Systems ◽

Density Functions ◽

Waiting Time Distribution ◽

Regenerative Processes ◽

Stieltjes Transform ◽

Expected Number ◽

Control Of Queues ◽

Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

Download Full-text

Stability of On-Line Bin Packing with Random Arrivals and Long-Run-Average Constraints

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800001753 ◽

1990 ◽

Vol 4 (4) ◽

pp. 447-460 ◽

Cited By ~ 7

Author(s):

Coastas Courcobetis ◽

Richard Weber

Keyword(s):

Bin Packing ◽

Random Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Expected Number ◽

Long Run ◽

Different Types ◽

On Line ◽

Necessary And Sufficient ◽

Uniformly Bounded

Items of various types arrive at a bin-packing facility according to random processes and are to be combined with other readily available items of different types and packed into bins using one of a number of possible packings. One might think of a manufacturing context in which randomly arriving subassemblies are to be combined with subassemblies from an existing inventory to assemble a variety of finished products. Packing must be done on-line; that is, as each item arrives, it must be allocated to a bin whose configuration of packing is fixed. Moreover, it is required that the packing be managed in such a way that the readily available items are consumed at predescribed rates, corresponding perhaps to optimal rates for manufacturing these items. At any moment, some number of bins will be partially full. In practice, it is important that the packing be managed so that the expected number of partially full bins remains uniformly bounded in time. We present a necessary and sufficient condition for this goal to be realized and describe an algorithm to achieve it.

Download Full-text

A note on the expected number of survivors in supercritical carrier-borne epidemics

Journal of Applied Probability ◽

10.2307/3213092 ◽

1979 ◽

Vol 16 (3) ◽

pp. 646-650 ◽

Cited By ~ 1

Author(s):

Roy Saunders ◽

Claude Lefèvre ◽

Richard J. Kryscio

Keyword(s):

Conditional Probability ◽

Epidemic Model ◽

Removal Rate ◽

Delta Function ◽

Formal Proof ◽

Initial Number ◽

Expected Number ◽

Kronecker Delta

We provide a formal proof of a conclusion due to Abakuks (1974) which states that the expected number of survivors in Downton's carrier-borne epidemic model approaches the limit (ρ /π)δ as the initial number of susceptibles tends to infinity. Here ρ denotes the relative removal rate for carriers, π denotes the conditional probability that an infected susceptible will become a carrier, δ denotes the Kronecker delta function and denotes the initial number of carriers.

Download Full-text