Handout # 8: Introduction to Queueing Theory

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Handout # 8: Introduction to Queueing Theory. CSC 2203 – Packet Switch and Network Architectures. Professor Yashar Ganjali Department of Computer Science University of Toronto [email protected] http://www.cs.toronto.edu/~yganjali Thanks to Monia Ghobadi.

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### Handout # 8:Introduction to Queueing Theory

CSC 2203 – Packet Switch and Network Architectures

Professor Yashar Ganjali

Department of Computer Science

University of Toronto

[email protected]

http://www.cs.toronto.edu/~yganjali

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A Simple Queue Model

One server, infinite number of waiting positions

Arrival rate: l,

∆t0, P{one arrival in time interval ∆t} = l ∆t

P{more than one arrival in ∆t} ~ 0 (negligible)

Average service rate: m

∆t0, P{one departure from the system in ∆t} = m ∆t

P{more than one departures from the system ∆t} ~ 0 (negligible)

Arrivals

Departures

Waiting

positions

Server(s)

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Markov Process
• Markov property: Memoryless
• For a stochastic process X(t) and any choice of time instants ti, i=1,…,n, we have
• P{X(tn+1)=xn+1|X(tn)=xn……. X(t1)=x1}=

P{X(tn+1)=xn+1|X(tn)=xn}

• The state of the process/system at time instant tn+1 depends only on the state of the process/system at the previous instant tn and not on any of the earlier time instants.
• Markov process: Give the present of state of the process, its future evolution is independent of the past of the process (one-step dependency feature).

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State Transition Diagram

Arrivals

Departures

Waiting

positions

Server(s)

No arrival

no departure

arrival

arrival

No arrival

no departure

No arrival

no departure

k-1

k

k+1

departure

departure

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The system state at any time instant may be taken as the number in the system at that instant.

pN(t) = P{system in state N at time t}

p0(t+ ∆t) = p0(t) [1-l∆t] + p1(t) m∆tN=0

pN(t+ ∆t) = pN(t) [1-l∆t-m ∆t] + pN-1(t) l∆t + pN+1(t)m ∆tN>0

Subject to the normalisation condition that:

∑ipi(t) = 1 for all t ≥0

Take limit as ∆t0

System State

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Equilibrium Solution
• These differential equations along with the normalization condition may be used to get the equilibrium solutions.
• The conditions invoked are:
• Define ρ=λ/μ, with ρ < 1 for stability we get:
• p1= ρ p0 (eq. 1)
• pN+1= (1+ρ) pN - ρ pN-1= ρpN=ρN+1 p0 N≥1

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System State Probabilities
• Solving eq.1 we get the system state probabilities:
• pi= ρi(1- ρ) i=0,1,……
• Note: The summation in normalization condition would only have a finite value when ρ<1.
• This condition is therefore required for the queue to be stable.
• Once we know the equilibrium state probabilities, we can use them to compute various mean performance parameters for this simple queue.

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Performance Parameters
• Mean number in system, N
• Mean number waiting in queue, Nq
• Mean time spent in system, W
• Mean time spent waiting in queue, Wq

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Lessons Learned
• The basic approach to the analysis of simple queuing models would begin by defining an appropriate system state for the queue.
• The analysis of queue would then essentially be the study of the way this system state would evolve.
• Interested in the performance analysis of the queue once equilibrium conditions have been reached.
• Review some of the basics of the theory of Markov Chains.

University of Toronto – Fall 2012