Waiting Times in BMAP/BMAP/1 Queues MAM-9, Budapest Nail Akar, - PowerPoint PPT Presentation

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Waiting Times in BMAP/BMAP/1 Queues MAM-9, Budapest Nail Akar, Bilkent University, Ankara, Turkey June 29, 2016 Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Table of Contents 1 Continuous-Valued Lindley Process 2 Markov Renewal Processes as Arrival and Service Models ME Distributions MRP MRP-ME MRP-ME Examples BMAP as an MRP-ME 3 Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Algorithm 4 Conclusions and Future Work Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Continuous-Valued Lindley Process Customers n n+1 Arriving Time A n n-1 n Customers W n B n Leaving W n+1 Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Lindley Equation for Waiting Times W n +1 = ( W n + B n − A n ) + = max (0 , W n + B n − A n ) , n ≥ 0 Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Lindley Equation for Waiting Times W n +1 = ( W n + B n − A n ) + = max (0 , W n + B n − A n ) , n ≥ 0 A n (interarrivals) and B n (service times) are very general Markov Renewal Processes with Matrix Exponential kernels, called an MRP-ME process. Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Lindley Equation for Waiting Times W n +1 = ( W n + B n − A n ) + = max (0 , W n + B n − A n ) , n ≥ 0 A n (interarrivals) and B n (service times) are very general Markov Renewal Processes with Matrix Exponential kernels, called an MRP-ME process. Goal: Obtain the steady-state distribution of W = lim n →∞ W n : F W ( t ) = Pr { W ≤ t } , f W ( t ) = F ′ W ( t ) Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

Continuous-Valued Lindley Process Markov Renewal Processes as Arrival and Service Models Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues Conclusions and Future Work Lindley Equation for Waiting Times W n +1 = ( W n + B n − A n ) + = max (0 , W n + B n − A n ) , n ≥ 0 A n (interarrivals) and B n (service times) are very general Markov Renewal Processes with Matrix Exponential kernels, called an MRP-ME process. Goal: Obtain the steady-state distribution of W = lim n →∞ W n : F W ( t ) = Pr { W ≤ t } , f W ( t ) = F ′ W ( t ) ρ = E [ B ] / E [ A ] < 1 Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function v ( h ) is a row (column) vector Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function v ( h ) is a row (column) vector T is a square matrix of size m Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function v ( h ) is a row (column) vector T is a square matrix of size m d = 1 + vT − 1 h is the probability mass at zero. Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function v ( h ) is a row (column) vector T is a square matrix of size m d = 1 + vT − 1 h is the probability mass at zero. The MGF g X ( s ) = E [ e − sX ] is rational � ∞ 0 − e − sx f X ( x ) dx = v ( s I − T ) − 1 h + d g X ( s ) = (2) Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME ME Distribution The non-negative random variable X ∼ ME( v , T , h , d ) has a PDF f X ( x ) f X ( x ) = ve Tx h + d δ ( x ) (1) where δ ( · ) denotes the dirac-delta function v ( h ) is a row (column) vector T is a square matrix of size m d = 1 + vT − 1 h is the probability mass at zero. The MGF g X ( s ) = E [ e − sX ] is rational � ∞ 0 − e − sx f X ( x ) dx = v ( s I − T ) − 1 h + d g X ( s ) = (2) E [ X i ] = ( − 1) i +1 i ! vT − ( i +1) h Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME Markov Renewal Process (MRP) X k ∈ { 1 , 2 , . . . , n } (modulating chain) Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME Markov Renewal Process (MRP) X k ∈ { 1 , 2 , . . . , n } (modulating chain) T k ∈ [0 , ∞ ) , 0 = T 0 ≤ T 1 ≤ T 2 ≤ · · · (arrival epochs) Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME Markov Renewal Process (MRP) X k ∈ { 1 , 2 , . . . , n } (modulating chain) T k ∈ [0 , ∞ ) , 0 = T 0 ≤ T 1 ≤ T 2 ≤ · · · (arrival epochs) ∆ k = T k +1 − T k (interarrival times: modulated process) P { X k +1 = j , T k +1 − T k ≤ t | X 0 , · · · , X k = i ; T 0 , . . . , T k } = P { X k +1 = j , T k +1 − T k ≤ t | X k = i } = F ij ( t ) Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues

ME Distributions Continuous-Valued Lindley Process MRP Markov Renewal Processes as Arrival and Service Models MRP-ME Steady-state Waiting Time in MRP-ME/MRP-ME/1 Queues MRP-ME Examples Conclusions and Future Work BMAP as an MRP-ME Markov Renewal Process (MRP) X k ∈ { 1 , 2 , . . . , n } (modulating chain) T k ∈ [0 , ∞ ) , 0 = T 0 ≤ T 1 ≤ T 2 ≤ · · · (arrival epochs) ∆ k = T k +1 − T k (interarrival times: modulated process) P { X k +1 = j , T k +1 − T k ≤ t | X 0 , · · · , X k = i ; T 0 , . . . , T k } = P { X k +1 = j , T k +1 − T k ≤ t | X k = i } = F ij ( t ) Semi-Markov Kernel F ( t ) = { F ij ( t ) } Nail Akar, Bilkent University, Ankara, Turkey Waiting Times in BMAP/BMAP/1 Queues