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Time-Dependent Infinite-Server General Queueing Networks. Michael R. Taaffe University of Minnesota November 12, 2001 OR590 OPERATIONS RESEARCH COLLOQUIUM SERIES Penn State University. umn.edu/~taaffe. Time-Dependent Processes. Managing Time-Dependent Processes.
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Time-Dependent Infinite-Server General Queueing Networks Michael R. Taaffe University of Minnesota November 12, 2001 OR590 OPERATIONSRESEARCH COLLOQUIUM SERIES Penn State University umn.edu/~taaffe
Time-dependent Ph-type queues • Time-dependent Pht arrival process • Time-dependent Pht service process • Time-dependent Markovian routing
Time-dependent Ph-type queues - Today • Time-dependent Pht arrival process • Time-dependent Pht service process • Time-dependent Markovian routing
E[# busy servers] Mt / Mt / ∞Moment Dif’l Equations (MDE’s) System Departure rate System Arrival Rate Closed set of MDE’s!
E[# busy servers] Mt / Mt / 1Moment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Not Closed!
Mt / Mt / 1Moment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Pseudo-Closed!
E[# busy servers] Mt / Mt / 1 / KMoment Dif’l Equations (MDE’s) System Arrival Rate System Departure rate Pseudo-Closed!
E[# busy servers] Mt / Mt / C / CMoment Diffr’l Equations (MDE’s) System Departure rate System Arrival Rate Pseudo-Closed!
Analysis Problems withNon-Exponential • Simple, first-order, linear differential equations have “no memory” (exponential) • “General” distributions have memory! • Not clear how to represent or interpret time-dependent “general” distributions
Initial-state distribution: New initial-state distribution: state-occupation times … … … … … … jumps around guided by A until absorbed Ph-type process … Markov the lion … … … Absorbing state
… … … … … … jumps around guided by A until absorbed Initial-state distribution: Ph-type process … Markov the lion … … … Absorbing state
New initial-state distribution: Initial-state distribution: state-occupation times … … … … … … jumps around guided by A(t) until absorbed Pht-type process … Markov the lion … … … Absorbing state
The Pht / Pht / ∞ } -The Random Process
Moment Differential Equations (MDE’s) Not closed!
Partial-Moment Differential Equations (PMDE’s) Not closed!
{{First PMDE’s} U {ADE’s}} Closed!
Higher-Order PMDE’s Not closed!
2cdPMDE’s Not closed!
The Pht / Pht / ∞CPMDE’s Not closed!
Pht / Pht / ∞ {ADE’s U 1st PMDE’s U CPMDE’s} is Closed! Quasi-Closure!
Pht / Pht / ∞ MAPLE Code www.iems.nwu.edu/~nelsonb/PhPhInf
The [Pht / Pht / ∞]KNetworkMAPLE Code www.iems.nwu.edu/~nelsonb/PhPhInf
cdf of the time spent in the system for entities arriving at time t