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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

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

Thanks to Monia Ghobadi

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A Simple Queue Model

One server, infinite number of waiting positions

Arrival rate: l,

∆t0, P{one arrival in time interval ∆t} = l ∆t

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

Average service rate: m

∆t0, 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)

University of Toronto – Fall 2012

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).

University of Toronto – Fall 2012

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

University of Toronto – Fall 2012

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 ∆t0

System StateUniversity of Toronto – Fall 2012

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

University of Toronto – Fall 2012

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.

University of Toronto – Fall 2012

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

University of Toronto – Fall 2012

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

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