Chapter 3 – Queueing Systems in Equilibrium

1. Chapter 3 – Queueing Systems in Equilibrium Leonard Kleinrock, Queueing Systems, Vol I: Theory Nelson Fonseca State University of Campinas, Brazil

2. 3. Birth-Death Queuing Systems in Equilibrium Equations: Transient solution Solution in equilibrium can not be generalized

3. 3.1. Solution in Equilibrium Whereas pk is no longer a function of t, we are not claiming that the process does not move from state to state in this limiting case The long-run probability of finding the system with k members will be properly described by pk

4. 0 1 k-1 k0 0 1 2 k-1 k k+1 m1 m2 mk mk+1 These equations are identical to the differential ones Flow rate into Ek = k-1pk-1 + k+1pk+1 Flow rate out of Ek = (k + k)pk In equilibrium (flow in = flow out)

5. Rather than surrounding each state we could choose a sequence of boundaries the first of which surrounds E0, the second of which surrounds E0 and E1, and so on, we would have the following relationship:

7. Is there a solution in equilibrium? Ergodic: S1 <  S2 =  Recurrent null: S1 =  S2 =  Transient: S1 =  S2 < 

8.     0 1 2 k-1 k k+1 m m m m The M|M|1 queue:

9. Using: we have:

12. 1-r (1-r) r pk (1-r) r2 (1-r) r3 0 1 2 3 4 5 k 

13. 0 1 r  0 1 r 

14. Probability of exceeding  geometrically decreasing

15. a a/2 a/k a/(k+1) 0 1 2 k-1 k k+1 m m m m 3.3. Discouraged Arrivals

17. l l l l 0 1 2 k-1 k k+1 m 2m km (k+1)m • 3.4. M|M|∞ (Infinite Server)

19. 3.5. M|M|m (mServer)

21. l l l l l l m+1 0 1 2 m-2 m-1 m m 2m (m-1)m mm mm mm

22. P[queueing] – probability that no server is available in a system of m servers. Erlang’s C formula C(m,l/m)

23. l l l l 0 1 2 K-2 K-1 K m m m m 3.6. M|M|1|kFinite Storage K=1 - Blocked calls cleared

26. 3.7. M|M|m|mm-Server Loss System

27. l l l l 0 1 2 m-2 m-1 m m 2m (m-1)m mm

28. 3.8. M|M|1||M

29. Ml (M-1)l 2l l 0 1 2 M-2 M-1 M m m m m

30. 3.9. M|M|∞||M

31. Ml (M-1)l 2l l 0 1 2 M-2 M-1 M m 2m (M-1)m Mm

32. 3.10. M|M|m|K|M

34. Problem 3.2 Consider a Markovian queueing system in which • Find the equilibrium probability pk of having k customers in the system. Express your answer in terms of p0. • Give an expression for p0.

35. Solution

36. So Note for 0≤a<1, this system is always stable.

37. Problem 3.5 Consider a birth-death system with the following birth and death coefficients: • Solve for pk. Be sure to express your answers explicitly in terms of l, k, and m only. • Find the average number of customers in the system.

38. Solution Here we demonstrate the “differentiation trick” for summing series (similar to that on page 69).

39. Since we have

40. Thus and so

41. (b)