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Queueing Network and Some Types of Customers and Signals Queueing Network and Some Types of Customers and Signals Thu-Ha Dao-Thi Chargee´ de recherche CNRS, PRiSM, UVSQ, France Joint work with J.M Fourneau, J. Mairesse, M.A Tran VMS-SMF Joint Congress, Hue, August 23th 2012 Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals Plan 1 Introduction-Preliminaries Queue? Network? 2 0-automatic queues and networks Introduction of 0-automatic queues Results on 0-automatic queues and networks 3 Some new types of signals in G-networks Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals Plan 1 Introduction-Preliminaries Queue? Network? 2 0-automatic queues and networks Introduction of 0-automatic queues Results on 0-automatic queues and networks 3 Some new types of signals in G-networks Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals Plan 1 Introduction-Preliminaries Queue? Network? 2 0-automatic queues and networks Introduction of 0-automatic queues Results on 0-automatic queues and networks 3 Some new types of signals in G-networks Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Outline 1 Introduction-Preliminaries Queue? Network? 2 0-automatic queues and networks Introduction of 0-automatic queues Results on 0-automatic queues and networks 3 Some new types of signals in G-networks Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals In daily life Dao Ha Queueing Network and Some Types of Customers and Signals Servers 1 the arriving process 2 . the discipline of service n Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study A.K. Erlang (1909): queue of telephones n : number of servers K : capacity of the buffer D : discipline of service FirstInFirstOut, LIFO, Kendall nomenclature (1953): PS, . A=S=n=K =D A : inter-arrival time distribution S : service time distribution Dao Ha Queueing Network and Some Types of Customers and Signals Servers 1 the arriving process 2 . the discipline of service n Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study A.K. Erlang (1909): queue of telephones n : number of servers K : capacity of the buffer D : discipline of service FirstInFirstOut, LIFO, Kendall nomenclature (1953): PS, . A=S=n=K =D A : inter-arrival time distribution S : service time distribution Dao Ha Queueing Network and Some Types of Customers and Signals Servers 1 the arriving process 2 . the discipline of service n Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study A.K. Erlang (1909): queue of telephones n : number of servers K : capacity of the buffer D : discipline of service FirstInFirstOut, LIFO, Kendall nomenclature (1953): PS, . A=S=n=K =D A : inter-arrival time distribution S : service time distribution Dao Ha Queueing Network and Some Types of Customers and Signals Servers 1 the arriving process 2 . the discipline of service n Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study A.K. Erlang (1909): queue of telephones n : number of servers K : capacity of the buffer D : discipline of service FirstInFirstOut, LIFO, Kendall nomenclature (1953): PS, . A=S=n=K =D A : inter-arrival time distribution S : service time distribution Dao Ha Queueing Network and Some Types of Customers and Signals p q Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Network Queues can be connected to form queueing networks Jackson network Kelly network Probabilistic routing fixed routing Dao Ha Queueing Network and Some Types of Customers and Signals p q Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Network Queues can be connected to form queueing networks Jackson network Kelly network Probabilistic routing fixed routing Dao Ha Queueing Network and Some Types of Customers and Signals p q Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Network Queues can be connected to form queueing networks Jackson network Kelly network Probabilistic routing fixed routing Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Application Dao Ha Queueing Network and Some Types of Customers and Signals 0-automatic network (Dao Thi and Mairesse) Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Product form solution Product form: a network () independent queues Jackson network (1957), BCMP network (Baskett et al, 1975), Kelly network (1979), G-network (Gelenbe, 1989) Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Product form solution Product form: a network () independent queues Jackson network (1957), BCMP network (Baskett et al, 1975), Kelly network (1979), G-network (Gelenbe, 1989) 0-automatic network (Dao Thi and Mairesse) Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Network with signals 90s: Negative customer (Gelenbe) Some types of signals: Reset, catastrophe, batch (Gelenbe, 1993, 2002, Chao, 1995) Negative signal, positive signal (Chao et al, 1999) Service time: exponential, Cox Recent results (Dao Thi, Fourneau and Tran, 2010, 2011, 2012): New types of signals: change class, group-deletion signal,... Service PH: signal change phase Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Network with signals 90s: Negative customer (Gelenbe) Some types of signals: Reset, catastrophe, batch (Gelenbe, 1993, 2002, Chao, 1995) Negative signal, positive signal (Chao et al, 1999) Service time: exponential, Cox Recent results (Dao Thi, Fourneau and Tran, 2010, 2011, 2012): New types of signals: change class, group-deletion signal,... Service PH: signal change phase Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries Introduction of 0-automatic queues 0-automatic queues and networks Results on 0-automatic queues and networks New type of signals Outline 1 Introduction-Preliminaries Queue? Network? 2 0-automatic queues and networks Introduction of 0-automatic queues Results on 0-automatic queues and networks 3 Some new types of signals in G-networks Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries Introduction of 0-automatic queues 0-automatic queues and networks Results on 0-automatic queues and networks New type of signals Two first examples of (0-automatic) queues The simple queue M=M=1=1=FIFO −!λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n 2 N µ µ ! + An arrival: n n 1 Burke theorem: departure Stability condition: λ < µ process is Poisson of rate λ ! a trivial random walk (r.w.) on (N; +), jumps +1. Gelenbe ’s G-queue with positive and negative customers 2 types of customers: f1g; {−1g. Buffer content: n 2 Z f1g-customer: n ! n + 1, {−1g-customer: n ! n − 1 ν: probability measure on {−1; 1g. ! a r.w. on (Z; +), jumps ±1 according to ν Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries Introduction of 0-automatic queues 0-automatic queues and networks Results on 0-automatic queues and networks New type of signals Two first examples of (0-automatic) queues The simple queue M=M=1=1=FIFO −!λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n 2 N µ µ ! + An arrival: n n 1 Burke theorem: departure Stability condition: λ < µ process is Poisson of rate λ ! a trivial random walk (r.w.) on (N; +), jumps +1. Gelenbe ’s G-queue with positive and negative customers 2 types of customers: f1g; {−1g. Buffer content: n 2 Z f1g-customer: n ! n + 1, {−1g-customer: n ! n − 1 ν: probability measure on {−1; 1g. ! a r.w. on (Z; +), jumps ±1 according to ν Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries Introduction of 0-automatic queues 0-automatic queues and networks Results on 0-automatic queues and networks New type of signals Two first examples of (0-automatic) queues The simple queue M=M=1=1=FIFO −!λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n 2 N µ µ ! + An arrival: n n 1 Burke theorem: departure Stability condition: λ < µ process is Poisson of rate λ ! a trivial random walk (r.w.) on (N; +), jumps +1. Gelenbe ’s G-queue with positive and negative customers 2 types of customers: f1g; {−1g. Buffer content: n 2 Z f1g-customer: n ! n + 1, {−1g-customer: n ! n − 1 ν: probability measure on {−1; 1g. ! a r.w. on (Z; +), jumps ±1 according to ν Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries Introduction of 0-automatic queues 0-automatic queues and networks Results on 0-automatic queues and networks New type of signals Two first examples of (0-automatic) queues The simple queue M=M=1=1=FIFO −!λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n 2 N µ µ ! + An arrival: n n 1 Burke theorem: departure Stability condition: λ < µ process is Poisson of rate λ ! a trivial random walk (r.w.) on (N; +), jumps +1.
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