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 Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study
A.K. Erlang (1909): queue of telephones
Servers n : number of 1 servers
the arriving process 2 K : capacity of . the buffer . the discipline of service D : discipline of n 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 Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study
A.K. Erlang (1909): queue of telephones
Servers n : number of 1 servers
the arriving process 2 K : capacity of . the buffer . the discipline of service D : discipline of n 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 Introduction-Preliminaries 0-automatic queues and networks Queue? Network? New type of signals Mathematical study
A.K. Erlang (1909): queue of telephones
Servers n : number of 1 servers
the arriving process 2 K : capacity of . the buffer . the discipline of service D : discipline of n 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 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
p
q
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
Queues can be connected to form queueing networks
Jackson network Kelly network Probabilistic routing fixed routing
p
q
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/∞/FIFO
−→λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n ∈ 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: {1}, {−1}. Buffer content: n ∈ Z {1}-customer: n → n + 1, {−1}-customer: n → n − 1 ν: probability measure on {−1, 1}. → 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/∞/FIFO
−→λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n ∈ 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: {1}, {−1}. Buffer content: n ∈ Z {1}-customer: n → n + 1, {−1}-customer: n → n − 1 ν: probability measure on {−1, 1}. → 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/∞/FIFO
−→λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n ∈ 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: {1}, {−1}. Buffer content: n ∈ Z {1}-customer: n → n + 1, {−1}-customer: n → n − 1 ν: probability measure on {−1, 1}. → 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/∞/FIFO
−→λ µ Stationary distribution: λ λ n π(n) = 1 − Buffer content: n ∈ 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: {1}, {−1}. Buffer content: n ∈ Z {1}-customer: n → n + 1, {−1}-customer: n → n − 1 ν: probability measure on {−1, 1}. → 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 Combine 2 models
Set of possible types: Σ = {a, b, b−1}. Buffer content: a word in ∗ −1 −1 L = uk ··· u1 ∈ Σ | ∀i, ui+1ui 6∈ {bb , b b} . (1) Customers of type a, b, b−1 are resp. in black, red and white.
ν: probability measure on Σ. → buffer content evolves as a Markov chain on L, ∗ a r.w. induced by ν on {a} ? F(b)
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 Combine 2 models
Set of possible types: Σ = {a, b, b−1}. Buffer content: a word in ∗ −1 −1 L = uk ··· u1 ∈ Σ | ∀i, ui+1ui 6∈ {bb , b b} . (1) Customers of type a, b, b−1 are resp. in black, red and white.
ν: probability measure on Σ. → buffer content evolves as a Markov chain on L, ∗ a r.w. induced by ν on {a} ? F(b)
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 4 types of “tasks”
Classical type.
Positive/negative type.
“One equals many” type.
“Dating agency” type.
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 4 types of “tasks”
Classical type.
Positive/negative type.
“One equals many” type.
“Dating agency” type.
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 4 types of “tasks”
Classical type.
Positive/negative type.
“One equals many” type.
“Dating agency” type.
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 4 types of “tasks”
Classical type.
Positive/negative type.
“One equals many” type.
“Dating agency” type. 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 Result for 0-automatic queues
For FIFO 0-automatic queues: Stationary distribution and stationary condition Burke theorem for departure process: Poisson −→ Consider the Jackson-like and Kelly-like network of 0-automatic queues
Dao-Thi and Mairesse, Adv. in Appl. Probab., 39(2) 2007.
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 Result for 0-automatic queues
For FIFO 0-automatic queues: Stationary distribution and stationary condition Burke theorem for departure process: Poisson −→ Consider the Jackson-like and Kelly-like network of 0-automatic queues
Dao-Thi and Mairesse, Adv. in Appl. Probab., 39(2) 2007.
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 Results on 0-automatic networks
Jackson-like network: Kelly-like network : fixed Routing matrix P = (pij )ij routing Fusion case b ∗ a = d : p 11 d1 p12 1 2
λ1 2
p13 1 4
b*a =d 3 3
Q i i i Product-form solution: π(u, α) = i π (u , α )
Dao-Thi and Mairesse, Discrete Event Dynamic Systems, V 18, N4, 2008
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 Results on 0-automatic networks
Jackson-like network: Kelly-like network : fixed Routing matrix P = (pij )ij routing Fusion case b ∗ a = d : p 11 d1 p12 1 2
λ1 2
p13 1 4
b*a =d 3 3
Q i i i Product-form solution: π(u, α) = i π (u , α )
Dao-Thi and Mairesse, Discrete Event Dynamic Systems, V 18, N4, 2008
Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 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 0-automatic queues and networks New type of signals Synchronised arrivals and group-deletion signals
Positive signal (X.Chao et al, 1999) : add a customer to a queue Synchronised arrivals: a signal will add one customer to some queues in the network Dao Thi, Fourneau and Tran, Performance Evaluation, V68, 2011
Group-deletion signals: delete all customer of the same class at the back-end of the buffer Dao Thi, Fourneau and Tran, EPEW 2011
Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals Synchronised arrivals and group-deletion signals
Positive signal (X.Chao et al, 1999) : add a customer to a queue Synchronised arrivals: a signal will add one customer to some queues in the network Dao Thi, Fourneau and Tran, Performance Evaluation, V68, 2011
Group-deletion signals: delete all customer of the same class at the back-end of the buffer Dao Thi, Fourneau and Tran, EPEW 2011
Dao Ha Queueing Network and Some Types of Customers and Signals φ(n) γ(l,n) l n l+1ll−1 n−1 l l−1 Departure ...... l+1 l l−1 n l+1 l−1
Symmetric discipline γ (l,n)
n−1 ll−1n l+1 l l−1 ... Arrival ...... l l−1 n−1 l l−1
Introduction-Preliminaries 0-automatic queues and networks New type of signals Change-class signal and change phase signal
Change-class Signal: Signal Sa,b: a-customer ,→ b-customer Sa,b ∗ a = b −→ link with 0-automatic mechanism Dao Thi, Fourneau and Tran, ASMTA 2010 Change-phase Signal: Phase-type service time Changing-phase signal: skip a phase of service Dao, Fourneau, Tran, The Computer Journal, V54, 2011
Dao Ha Queueing Network and Some Types of Customers and Signals φ(n) γ(l,n) l n l+1ll−1 n−1 l l−1 Departure ...... l+1 l l−1 n l+1 l−1
Symmetric discipline γ (l,n)
n−1 ll−1n l+1 l l−1 ... Arrival ...... l l−1 n−1 l l−1
Introduction-Preliminaries 0-automatic queues and networks New type of signals Change-class signal and change phase signal
Change-class Signal: Signal Sa,b: a-customer ,→ b-customer Sa,b ∗ a = b −→ link with 0-automatic mechanism Dao Thi, Fourneau and Tran, ASMTA 2010 Change-phase Signal: Phase-type service time Changing-phase signal: skip a phase of service Dao, Fourneau, Tran, The Computer Journal, V54, 2011
Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals Change-class signal and change phase signal
Change-class Signal: Signal Sa,b: a-customer ,→ b-customer Sa,b ∗ a = b −→ link with 0-automatic mechanism Dao Thi, Fourneau and Tran, ASMTA 2010 Change-phase Signal: Phase-type service time Changing-phase signal: skip a phase of service Dao, Fourneau, Tran, The Computer Journal, V54, 2011
φ(n) γ(l,n) l n l+1ll−1 n−1 l l−1 Departure ...... l+1 l l−1 n l+1 l−1
Symmetric discipline γ (l,n)
n−1 ll−1n l+1 l l−1 ... Arrival ...... l l−1 n−1 l l−1 Dao Ha Queueing Network and Some Types of Customers and Signals Introduction-Preliminaries 0-automatic queues and networks New type of signals
Dao Ha Queueing Network and Some Types of Customers and Signals