Queues with vacations and their applications Dieter Fiems and Herwig - PowerPoint PPT Presentation

Queues with vacations and their applications Dieter Fiems and Herwig Bruneel SMACS Research Group, Ghent University, Belgium { df,hb } @UGent.be IPS-MoMe, Warsaw, Poland – p.1/26

Outline • Vacations • Mathematical model • Queueing analysis • Special cases • Case study: Priority queues • Conclusions IPS-MoMe, Warsaw, Poland – p.2/26

Vacations – What? • Queueing theory parlance for temporary server unavailability • Resource sharing • Breakdowns • Maintenance • Errors • Reconfiguration • ... • Correlation structure? IPS-MoMe, Warsaw, Poland – p.3/26

Vacations – Resource Sharing • Passive Optical Network Optical Network Unit 1 Optical Line Terminal Optical Network Unit 2 Optical Network Unit 3 IPS-MoMe, Warsaw, Poland – p.4/26

Vacations – Resource Sharing • Passive Optical Network Optical Network Unit 1 Optical Line Terminal Optical Network Unit 2 Vantage point ONU 1 Optical Network Unit 3 ONU 2 ONU 1 ONU 3 ONU 2 time Available Vacation IPS-MoMe, Warsaw, Poland – p.4/26

Vacations – Resource Sharing • Ethernet ... Bus IPS-MoMe, Warsaw, Poland – p.5/26

Vacations – Resource Sharing • Ethernet ... Bus Vacation Back−off Station 1 time Back−off Station 2 time Collision Vacation IPS-MoMe, Warsaw, Poland – p.5/26

Vacations – Resource Sharing • Service differentiation Class 1 Class 2 IPS-MoMe, Warsaw, Poland – p.6/26

Vacations – Resource Sharing • Service differentiation Class 1 Class 2 • Priority Queueing • Weighted Round Robin • Weighted Fair Queueing IPS-MoMe, Warsaw, Poland – p.6/26

✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✁ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✁ ✄ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ IPS-MoMe, Warsaw, Poland – p.7/26 Vacations – Errors • Go-Back-N ARQ Errors

✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✆ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✝ ✞ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✝ ✆ ✞ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ☎ ✆ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✞ IPS-MoMe, Warsaw, Poland – p.7/26 time time ACK Vacation Error R S Vacations – Errors • Go-Back-N ARQ Errors

Vacations – Non-telecom Unsignalised intersection IPS-MoMe, Warsaw, Poland – p.8/26

Vacations – Non-telecom Unsignalised intersection Airplane queue IPS-MoMe, Warsaw, Poland – p.8/26

Vacations – Models • Desirable properties of vacation process • Correlation between vacations • Service interruptions • Other dependences IPS-MoMe, Warsaw, Poland – p.9/26

Vacations – Models • Desirable properties of vacation process • Correlation between vacations • Service interruptions • Other dependences • Desirable properties of queueing system • Realistic arrival process • General service times IPS-MoMe, Warsaw, Poland – p.9/26

Vacations – Models • Desirable properties of vacation process • Correlation between vacations • Service interruptions • Other dependences • Desirable properties of queueing system • Realistic arrival process • General service times • Approaches • Analytic methods • Numerical methods • Simulation IPS-MoMe, Warsaw, Poland – p.9/26

Vacations – Models • Desirable properties of vacation process • Correlation between vacations • Service interruptions • Other dependences • Desirable properties of queueing system • Realistic arrival process • General service times • Approaches • Analytic methods • Numerical methods • Simulation IPS-MoMe, Warsaw, Poland – p.9/26

Mathematical Model • Discrete-time queueing system synchronisation slot k slot k+1 time arrivals departure IPS-MoMe, Warsaw, Poland – p.10/26

Mathematical Model • Discrete-time queueing system synchronisation slot k slot k+1 time arrivals departure • Arrivals per slot, service times • independent and identically distributed • probability generating functions: E ( z ) and S ( z ) • service times are bounded IPS-MoMe, Warsaw, Poland – p.10/26

Mathematical Model • Infinite capacity queue IPS-MoMe, Warsaw, Poland – p.11/26

Mathematical Model • Infinite capacity queue • Single server system IPS-MoMe, Warsaw, Poland – p.11/26

Mathematical Model • Infinite capacity queue • Single server system • Vacation process state of the i j vacation of n slots ✲ vacation process ✲ ( k ) a during customer service  ❄  b customer leaves non-empty system queueing state c customer leaves empty system  d empty system Server in vacation state i and queueing state k takes a vacation of length n and goes to state j with probability b ( k ) ij ( n ) IPS-MoMe, Warsaw, Poland – p.11/26

Mathematical model • Dealing with interrupted service IPS-MoMe, Warsaw, Poland – p.12/26

Mathematical model • Dealing with interrupted service • Continue after interruption (CAI) 5 slots service time ✲ ✻ ❄ IPS-MoMe, Warsaw, Poland – p.12/26

Mathematical model • Dealing with interrupted service • Continue after interruption (CAI) 5 slots service time ✲ ✻ ❄ • Repeat after interruption (RAI) ✲ ✻ ❄ IPS-MoMe, Warsaw, Poland – p.12/26

Mathematical model • Dealing with interrupted service • Continue after interruption (CAI) 5 slots service time ✲ ✻ ❄ • Repeat after interruption (RAI) ✲ ✻ ❄ • Repeat after interruption with resampling (RAI,wr) service time resampled to 6 slots ✲ ✻ ❄ IPS-MoMe, Warsaw, Poland – p.12/26

Queueing Analysis • Probability generating functions approach • Matrices to deal with the finite state space of the vacation process IPS-MoMe, Warsaw, Poland – p.13/26

Queueing Analysis • Probability generating functions approach • Matrices to deal with the finite state space of the vacation process • Effective service times • Defined as: “the number of slots between the beginning of the slot where the customer is first served until the end of the slot where the customer leaves the system” • Effective service time analysis for the different operation modes • Unified queueing analysis IPS-MoMe, Warsaw, Poland – p.13/26

Queueing Analysis • Effective service time for CAI S T 1 2 3 S − 1 1 2 3 � T = 1 + Ω j time j =1 Ω 1 Ω 2 1 IPS-MoMe, Warsaw, Poland – p.14/26

Queueing Analysis • Effective service time for CAI S T 1 2 3 S − 1 1 2 3 � T = 1 + Ω j time j =1 Ω 1 Ω 2 1 ∞ � s ( n )Ω( z ) j − 1 Ω( z ) = B a ( z ) z T ( z ) = z n =1 IPS-MoMe, Warsaw, Poland – p.14/26

Queueing Analysis • Effective service time for RAI T S 1 2 3 1 2 1 1 2 3 Γ B T’ Γ + B + T ′ (int.) � T = S (no int.) IPS-MoMe, Warsaw, Poland – p.15/26

Queueing Analysis • Effective service time for RAI T S 1 2 3 1 2 1 1 2 3 Γ B T’ Γ + B + T ′ (int.) � T = S (no int.) ” − 1 “ X z ( z B a (0)) k − 1 I N − z [ zB a (0) − I N ] − 1 [( z B a (0)) k − 1 − I N ][ B a ( z ) − B a (0)] T ( z ) = s ( k ) k IPS-MoMe, Warsaw, Poland – p.15/26