Fluctuations of random walk in Rd and storage systems

Priscilla Greenwood; Moshe Shaked

doi:10.2307/1426115

Fluctuations of random walk in R d and storage systems

Advances in Applied Probability ◽

10.1017/s0001867800028986 ◽

1977 ◽

Vol 9 (03) ◽

pp. 566-587 ◽

Cited By ~ 3

Author(s):

Priscilla Greenwood ◽

Moshe Shaked

Keyword(s):

Random Walk ◽

Unique Solution ◽

Convex Cone ◽

Storage Systems ◽

Queueing Systems ◽

Convolution Equation ◽

Stopping Times ◽

Multivariate Distributions ◽

Explicit Formulas ◽

And Storage

Two Wiener-Hopf type factorization identities for multivariate distributions are introduced. Properties of associated stopping times are derived. The structure that produces one factorization also provides the unique solution of the Wiener-Hopf convolution equation on a convex cone in R d . Some applications for multivariate storage and queueing systems are indicated. For a few cases explicit formulas are obtained for the transforms of the associated stopping times. A result of Kemperman is extended.

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A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Journal of Applied Probability ◽

10.1239/jap/1378401226 ◽

2013 ◽

Vol 50 (3) ◽

pp. 612-631 ◽

Cited By ~ 2

Author(s):

D. Perry ◽

W. Stadje ◽

S. Zacks

Keyword(s):

Steady State ◽

Storage Systems ◽

Storage System ◽

Queueing Systems ◽

Local Maxima ◽

State Dependent ◽

State Densities ◽

Duality Approach ◽

And Storage ◽

Level Process

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

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A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Journal of Applied Probability ◽

10.1017/s0021900200009748 ◽

2013 ◽

Vol 50 (03) ◽

pp. 612-631 ◽

Cited By ~ 2

Author(s):

D. Perry ◽

W. Stadje ◽

S. Zacks

Keyword(s):

Steady State ◽

Storage Systems ◽

Storage System ◽

Queueing Systems ◽

Local Maxima ◽

State Dependent ◽

State Densities ◽

Duality Approach ◽

And Storage ◽

Level Process

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

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HOUSEHOLD WATER BALANCE AND ASSESSMENT OF WATER VOLUME FOR IRRIGATION IN AGRO LANDSCAPES OF THE KYZYLSU-YUZHNAYA BASIN

Prirodoobustrojstvo ◽

10.26897/1997-6011-2020-5-102-109 ◽

2020 ◽

pp. 102-109

Author(s):

D.KH. DOMULLODZHANOV ◽

◽

R. RAHMATILLOEV

Keyword(s):

Water Balance ◽

River Basin ◽

Water Use ◽

Storage Systems ◽

Low Cost ◽

Field Studies ◽

Water Volume ◽

Land Productivity ◽

Household Water ◽

And Storage

The article presents the results of the field studies and observations that carried out on the territory of the hilly, low-mountain and foothill agro landscapes of the Kyzylsu-yuzhnaya (Kyzylsu-Southern) River Basin of Tajikistan. Taking into account the high-altitude location of households and the amount of precipitation in the river basin, the annual volumes of water accumulated with the use of low-cost systems of collection and storage of precipitation have been clarified. The amount of water accumulated in the precipitation collection and storage systems has been established, the volume of water used for communal and domestic needs,the watering of livestock and the amount of water that can be used to irrigate crops in the have been determined. Possible areas of irrigation of household plots depending on the different availability of precipitation have been determined. It has been established that in wet years (with precipitation of about 10%) the amount of water collected using drip irrigation will be sufficient for irrigation of 0.13 hectares, and in dry years (with 90% of precipitation) it will be possible to irrigate only 0.03 ha of the household plot. On the basis of the basin, the total area of irrigation in wet years can be 4497 ha, and in dry years only 1087 ha. Taking into account the forecasts of population growth by 2030 and an increase in the number of households, the total area of irrigation of farmlands in wet years may reach 5703 hectares,and in dry years – 1379 hectares. Growing crops on household plots under irrigation contributes to a significant increase in land productivity and increases the efficiency of water use of the Kyzylsu-yuzhnaya basin.

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