Phase-type Distributions for Realistic Modelling in Discrete-Event - PowerPoint PPT Presentation

Phase-type Distributions for Realistic Modelling in Discrete-Event Simulation Philipp Reinecke and G´ abor Horv´ ath philipp.reinecke@fu-berlin.de hgabor@webspn.hit.bme.hu

Motivation: The Restart Method Restart: A client sends a request. If there is no response within a reasonable time, the request is repeated Restart may reduce response-times Question: When should the client restart the request? Small timeout → Low response-times, but also high additional system load Large timeout → Low additional load, but high response-times Application scenarios: Service-Oriented Systems (SOAs), WMNs, etc. What happens if everyone does it? 1 / 12

Evaluation Approaches Analysis � x F ( x ) = 0 f ( u ) du Simulation Experimental 2 / 12

Combined Approach Abstract methods give general results, but are often not realistic Practical methods are more realistic, but give less general results → Combine methods to obtain realistic and general results Requirements: Phenomena (e.g. response-times) must be modelled Models are required . . . must be accurate . . . must be fast . . . must be suitable for all abstraction levels Ideal models: Phase-type (PH) distributions. 3 / 12

Phase-type distributions λ 1 λ 2 λ 3 λ 4 A PH distribution is the distribution of the time to absorption in a Markov chain with one absorbing state Examples: Exponential distribution Hyperexponential distribution Erlang distribution Hypoexponential distribution 4 / 12

PH-Distributions for Modelling Use PH distributions to model delays, response-times, failure-times, etc. in test-beds, simulations, and abstract models Advantages over other distributions: Flexibility → Capture important system properties by fitting PH distributions to measurements Generic representations → Catch-all routines for random-variate generation Markovian representations → Suitable for analytical approaches Seldom used in simulation little-known difficult theory little to no support in simulators efficiency concerns 5 / 12

The Libphprng Library A library for generating random variates from PH distributions Part of the Butools collection http://webspn.hit.bme.hu/~butools Advantages: easy to use portable between simulators fast 6 / 12

Libphprng features Uniform Random RandomSourceWrapper Source BuToolsGenerator libphprng Core Simulation Code Shared library with small wrapper code for the uniform random number stream Application: 1 Create BuToolsGenerator object for the distribution 2 Register uniform random number stream 3 Draw random variates For other simulators: Write your own wrapper 7 / 12

Efficiency concerns λ 1 λ 2 λ 3 λ 4 Random-variate generation by ‘playing’ the Markov chain Costs depend on the structure and the algorithm . . . e.g. for a chain we do not need to randomly select the next state Structures are not unique Costs can be optimised by changing the structure Libphprng implements efficient algorithms and optimises the structure for random-variate generation 8 / 12

Evaluation Clients Server Jobs F S ( t ) Responses Evaluation of quality and performance Quality: Evaluation of restart timeouts Different models: cPSquare Exponential distribution Lognormal distribution Phase-type distribution (50 phases) 9 / 12

Evaluation Evaluation of quality and performance Quality: Evaluation of restart timeouts Different models: cPSquare Exponential distribution Lognormal distribution Phase-type distribution (50 phases) Performance: Simple source/sink model 9 / 12

Evaluation: Quality 6 Density 4 Empirical (Histogram) cPSquare Model Exponential Model Lognormal Model APH Model 2 0 0 1 2 3 4 5 6 7 Service time (s) 10 / 12

Evaluation: Quality 3 cPSquare Model Exponential Model Lognormal Model 2.5 APH Model Response-time (s) 2 1.5 1 0.5 0 0 1 2 3 4 5 Timeout (s) 10 / 12

Evaluation: Quality 3 cPSquare Model Exponential Model Lognormal Model 2.5 6 APH Model Response-time (s) 2 Density 4 Empirical (Histogram) 1.5 cPSquare Model Exponential Model Lognormal Model APH Model 1 2 0.5 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 Service time (s) Timeout (s) Not all models capture the density well Comparison of results: Only the PH model shows the existence of an optimal timeout 10 / 12

Evaluation: Performance 6e+06 5e+06 Simulation speed (ev/sec) 4e+06 3e+06 2e+06 1e+06 0 Exponential Lognormal libphprng ArrivalProcess 11 / 12

Evaluation: Performance 100 80 % of simulation time 60 40 20 0 Exponential Lognormal libphprng ArrivalProcess 11 / 12

Evaluation: Performance 6e+06 100 5e+06 80 Simulation speed (ev/sec) 4e+06 % of simulation time 60 3e+06 40 2e+06 20 1e+06 0 0 Exponential Lognormal libphprng ArrivalProcess Exponential Lognormal libphprng ArrivalProcess Libphprng is less efficient than the simpler models Libphprng is more efficient than ArrivalProcess by Kriege et al. (2011) . . . but only supports PH 11 / 12

Conclusion Libphprng enables accurate and efficient modelling of distributions in simulations using PH distributions Libphprng is portable between simulators Available from http://webspn.hit.bme.hu/~butools 12 / 12

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