The Asymmetric Inclusion Process (ASIP) is an n-site stochastic tandem network where single particles arrive randomly at the first site and move in clusters, simultaneously and unidirectionally, from one site to the next until exiting the system. Each site has a buffer of unlimited size and a gate in front of it that opens from time to time according to some specified stochastic rules. Various ASIP models, as well as generalized ASIP systems, were extensively investigated. Independently, queues in random environment were also studied in the queueing literature. Operating under ‘random environment’, the system as a whole alternates stochastically between several phases (environments), while residing in each phase a random time with a phase-specific characteristic. In this work, we consider a combined system–ASIP in random environment–where the system alternates between two environments, residing in phase k (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k = 1,2\)</EquationSource> </InlineEquation>) an exponentially distributed time with mean <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\frac{1}{\eta _k}\)</EquationSource> </InlineEquation>. When in environment k, single particles arrive at the first site following a Poisson process with rate <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\lambda _k\)</EquationSource> </InlineEquation>, and propagate in clusters unidirectionally through the network’s sites. The gate of site j (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(j = 1, 2, \ldots , n\)</EquationSource> </InlineEquation>) opens randomly, independently of the other gates, as follows: when the system is in phase k, gate j opens every exponentially distributed time with mean <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\frac{1}{\mu _{jk}}\)</EquationSource> </InlineEquation>, causing all particles accumulated in site j to move forward instantaneously and simultaneously, as a batch, to site <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(j+1\)</EquationSource> </InlineEquation> (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(j = 1, 2, \ldots , n-1\)</EquationSource> </InlineEquation>). When gate n opens, all particles present there exit the system. Moreover, at each change of phase, all particles present in the entire system are cleared (a disaster). We study this combined system – ASIP in random environment with disasters – and derive the following: (i) the multi-dimensional probability generating function (PGF) of site occupancies; (ii) the mean occupancy of each site; (iii) the probability that site j (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(j = 1, 2, \ldots , n\)</EquationSource> </InlineEquation>) is occupied; (iv) the mean sojourn time of a particle in the system; (v) the PGF and mean of the overall load of the first m sites (<InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(m = 1, 2,\ldots , n\)</EquationSource> </InlineEquation>); (vi) the mean duration of a busy period; (vii) the probability that the first occupied site is site m (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(m = 1, 2, \ldots , n\)</EquationSource> </InlineEquation>); and (viii) the mean length of the draining time (the time elapsing from the moment the arrival process stops until the first moment thereafter that the system becomes completely empty). The results are compared with those of the classical single-environment ASIP.

ASIP in random environment with disasters

期刊论文
发表日期：

2026年1月16日

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Yaron Yeger,
Uri Yechiali

摘要

The Asymmetric Inclusion Process (ASIP) is an n-site stochastic tandem network where single particles arrive randomly at the first site and move in clusters, simultaneously and unidirectionally, from one site to the next until exiting the system. Each site has a buffer of unlimited size and a gate in front of it that opens from time to time according to some specified stochastic rules. Various ASIP models, as well as generalized ASIP systems, were extensively investigated. Independently, queues in random environment were also studied in the queueing literature. Operating under ‘random environment’, the system as a whole alternates stochastically between several phases (environments), while residing in each phase a random time with a phase-specific characteristic. In this work, we consider a combined system–ASIP in random environment–where the system alternates between two environments, residing in phase k ( \(k = 1,2\) ) an exponentially distributed time with mean \(\frac{1}{\eta _k}\) . When in environment k, single particles arrive at the first site following a Poisson process with rate \(\lambda _k\) , and propagate in clusters unidirectionally through the network’s sites. The gate of site j ( \(j = 1, 2, \ldots , n\) ) opens randomly, independently of the other gates, as follows: when the system is in phase k, gate j opens every exponentially distributed time with mean \(\frac{1}{\mu _{jk}}\) , causing all particles accumulated in site j to move forward instantaneously and simultaneously, as a batch, to site \(j+1\) ( \(j = 1, 2, \ldots , n-1\) ). When gate n opens, all particles present there exit the system. Moreover, at each change of phase, all particles present in the entire system are cleared (a disaster). We study this combined system – ASIP in random environment with disasters – and derive the following: (i) the multi-dimensional probability generating function (PGF) of site occupancies; (ii) the mean occupancy of each site; (iii) the probability that site j ( \(j = 1, 2, \ldots , n\) ) is occupied; (iv) the mean sojourn time of a particle in the system; (v) the PGF and mean of the overall load of the first m sites ( \(m = 1, 2,\ldots , n\) ); (vi) the mean duration of a busy period; (vii) the probability that the first occupied site is site m ( \(m = 1, 2, \ldots , n\) ); and (viii) the mean length of the draining time (the time elapsing from the moment the arrival process stops until the first moment thereafter that the system becomes completely empty). The results are compared with those of the classical single-environment ASIP.