State behavior description

1. State behavior description • There are two ways to get the equations • Monitor the behavior of # of customers in a system • Subject to arrivals and departures • First way: Kolmogorov approach • That we studied last time • Second way: rate diagram • Key driver of today’s lecture

2. Pn (t) Steady state transient 1 t t+dt t ? n n-1: arrival n+1: departure n: none of the above Birth and death process: Kolmogorov approach N(t) = # of customers at time t. μn λn arrivals (births) departures (deaths)

3. Differential equation: steady state analysis • Limiting case

4. Complex example: 2 queues in tandem λ2 μ1 μ1 n1 n1 • State space • (n1, n2) • n1 = # customers in the first queue • N2 = # customers in the second queue • P(n1, n2) ? λ1 p 1-p

5. Kolmogorov approach • Think in terms of P(n1, n2)(t+dt) • In order to end up having (n1, n2) at time t+dt • Where do I need to be at time t • Moreover, what event would take place • To have n1 customers in queue 1, and n2 in queue 2 at t+dt t t+dt (n1, n2 ) (n1, n2 ) (1 – (λ1 +μ1+λ2 +μ2)dt) μ1dt (1-p) μ2dt λ1 dt λ2 dt (n1+1, n2 ) (n1, n2+1) (n1-1, n2 ) (n1, n2 -1) (n1+1, n2 -1) μ1.dt.p

6. Solution according to the classical approach • Limitation of the classical approach • Unmanageable when the problem • Gets more and more complicated

7. λ0 λ1 λ2 λn-1 λn μn μn+1 μ1 μ2 μ3 Rate diagram: simple problem • Classical approach (for the case of one queue) • Steady state analysis • Balance equation 0 1 2 3 n ….. …..

8. λn Rate of transition • Rate of transition from n to n+1 • Average # times the system moves from n to n+1 • => Average # of arrivals when we have n customers • => λn n n+1

9. 0,0 0,1 0,n2 1,1 1,0 1, n2 2,0 2, n2 2,1 3,0 3, n2 3,1 ….. ….. ….. λ1 λ1 λ1 λ1 λ1 λ1 λ1 λ1 λ1 Rate diagram: complicated problem • Consider rate diagram approach • For the case of the two queues . . . . . . . . . . . .

10. 0 1 2 3 n ….. ….. λ0 λ1 λ2 λn-1 λn μn μn+1 μ1 μ2 μ3 Balanced system • If the process is in equilibrium • => the average # times (per u.t.) • That the process enters a state n • Is equal to • The average # times • The process exits state n • These are called • Balance equations • Rate into a state = rate out of a state

11. Reversible Markov process • Solving the equations • These equations are called • Local balance equations • Balance specific flows • If a system satisfies these individual local equations • => Reversible Markov process • => it will have a product form solution

12. Z-transforms: generating functions • If we have a sequence of numbers {f0,f1 ,f2 , …,fk ,..} • It is often desirable to compress it into a single function • This process of converting a sequence of numbers • Into a single function is called the z-transformation • The resultant function is called the z-transform of numbers • The z-transform of a sequence is defined as

13. Z-transform: application in queuing systems • X is a discrete r.v. • P(X=i) = Pi, i=0, 1, … • P0 , P1 , P2 ,… • Properties of the z-transform • g(1) = 1, P0 = g(0); P1 = g’(0); P2 = ½ . g’’(0) • , +

14. Polynomial form