Handout # 8: Introduction to Queueing Theory

1. Handout # 8:Introduction to Queueing Theory CSC 2203 – Packet Switch and Network Architectures Professor Yashar Ganjali Department of Computer Science University of Toronto yganjali@cs.toronto.edu http://www.cs.toronto.edu/~yganjali Thanks to Monia Ghobadi TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

2. 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) University of Toronto – Fall 2012

3. 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

4. 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

5. 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 University of Toronto – Fall 2012

6. 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

7. 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

8. 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

9. 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