NATCOR: Stochastic Modelling

Transcription

1 NATCOR: Stochastic Modelling Queueing Theory II Chris Kirkbride Management Science 2017

2 Overview of Today s Sessions I Introduction to Queueing Modelling II Multiclass Queueing Models III Queueing Control IV Departure from Exponential Service Times V Time-dependent Behaviour of Queues VI Case Study: Time-dependent Queueing

3 Overview of this Session Multiclass M/M/1 queue Scheduling Wrap-up

4 Multiclass M/M/1 queue Arrivals λ 1 λ 2 λ J Queues SCHEDULING Service µ

5 Scheduling Policies FIFO/FCFS: serve in arrival order LIFO/LCFS: serve in reverse arrival order SPT: shortest processing time first SIRO: serve in random order PS: processor sharing PNPN: priority service, preemptive, non-preemptive Will will consider the first and last policies: M/M/1/ /FIFO M/M/1/ /PNPN

6 PASTA property PASTA property Poisson Arrivals See Time Averages: In steady-state, the system state seen by an arrival from a Poisson process has the same distribution as that seen by a random observer at a point in time. Intuition: State seen at t is impacted by arrivals before t Interarrival times are exponentially distributed (Memoryless) Interval between t and the next arrival has the same distribution whether t is an arrival instant or not

7 Utilization Law Utilization Law The probability that a customer of class i is being served is ρ i = λ i /µ i. Let δ i be the probability that class i is being served for i = 1,..., J and Consider W i = W qi + µ 1 i and L i = L qi + δ i. L i = λ i W i and L qi = λ i W qi.

8 Utilization Law Then L i L qi = λ i W i λ i W qi δ i = λ i /µ i = ρ i. Note: if ρ = ρ 1 + ρ ρ J < 1 then we have steady-state.

9 Multiclass M/M/1/ /FIFO queue Job from class i arrives. Its service is delayed by: Any ob receiving service W 0 = J ρ µ 1. Jobs awaiting service Hence J L q µ 1 = J ρ W q. W qi = J ρ µ 1 + J ρ W q

10 Multiclass M/M/1/ /FIFO queue Under FIFO W q1 =... = W qj = W q and so W q = J ρ µ 1 1 J ρ. Application of Little s Law provides W i, L i and L qi for i = 1,..., J.

11 Multiclass M/M/1/ /PNPN non-preemptive queue Priorities: J. Job from class i arrives. Its service is delayed by: Any ob receiving service W 0 = J ρ µ 1. Same or higher priority obs awaiting service i L q µ 1 = i ρ W q. Higher priority obs that arrive while waiting i 1 λ W qi µ 1 i 1 = W qi ρ.

12 Multiclass M/M/1/ /PNPN non-preemptive queue Hence, i i 1 W qi = W 0 + ρ W q + W qi ρ. The solution of which resolves to the Cobham equations W qi = W 0 (1 i 1 ρ )(1 i ρ ) = J ρ µ 1 (1 i 1 ρ )(1 i ρ ). Other class i performance measures W i, L i and L qi can be obtained via Little s Law.

13 Multiclass M/M/1/ /PNPN preemptive queue Priorities: J. Job from class i arrives. Jobs of lower priority are ignored (preempted by i) Service delayed by same/higher priority obs in service/queue (initial delay) Its service is interrupted by arrivals of higher priority obs (overall service time) Its service is initially delayed by: Ignore obs from class i + 1, i + 2,..., J Invariant of scheduling policy Wait equal to average wait of non-preemptive policy in which classes i + 1, i + 2,..., J don t exist Modified Cobham equations for class i are W I qi = W 0i (1 i 1 ρ )(1 i ρ ) ; W 0i = i ρ µ 1.

14 Multiclass M/M/1/ /PNPN preemptive queue Average overall service time for a class i ob, τ i, is impacted by preemptions from higher priority obs. Thus W i = W I qi + τ i = τ i = µ 1 = µ 1 i i 1 i + λ τ i µ 1 i 1 + τ i ρ. 1 W 0i (1 i 1 ρ )(1 i ρ ) + µ i 1 i 1 ρ Little s Law gives then provides W qi = W i µ 1 i, L i and L qi.

15 Wrap up Here, we have: Looked at multiclass queueing models Introduced scheduling policies Queueing models: Extending our models allows us to consider more realistic systems. Question: Is the implementation of a priority rule always of value over a simpler rule (FIFO, say)? Next: Optimal scheduling policies...