Queuing Systems: Lecture 3 Amedeo odoni October 17. 2001
Queuing Systems: Lecture 3 Amedeo R. Odoni October 17, 2001
Lecture outline M/M/1: finite system capacity, K M/MIm: finite system capacity, K M/M/m: finite system capacity, Kem Related observations and extensions M/E,/l example M/G/: epochs and transition probabilities M/G1: derivation of l Why M/G/m, G/G/1, etc are difficult
Lecture Outline • M/M/1: finite system capacity, K • M/M/m: finite system capacity, K • M/M/m: finite system capacity, K=m • Related observations and extensions • M/E2 /1 example • M/G/1: epochs and transition probabilities • M/G/1: derivation of L • Why M/G/m, G/G/1, etc. are difficult
M/M/1: finite system capacity, K customers finding system full are lost p"·(1-p) K+1 forn=0,1,2,…,k Steady state is always reached Be careful in applying Little's Law! Must count only the customers who actually join the system =:(1-Pk)
M/M/1: finite system capacity, K; customers finding system full are lost … 0 1 2 K-1 K l l l l l m m m m m P for n K K n n 0, 1, 2, ....., 1 (1 ) 1 = - × - = + r r r • Steady state is always reached • Be careful in applying Little’s Law! Must count only the customers who actually join the system: l¢ = l ×(1- PK )
M/M/m: finite system capacity, K; customers finding system full are lost m 2 Can write system balance equations and obtain closed form expressions for Pn, L, W, Lg, Wg Often useful in practice
M/M/m: finite system capacity, K; customers finding system full are lost 0 1 2 m m+1 K-1 l l l l l l 3m m 2m mm mm mm K l mm mm l …… …… • Can write system balance equations and obtain closed form expressions for Pn , L, W, Lq , Wq • Often useful in practice
M/M/m: finite system capacity, m special case 3 m 2u forn=0,1,2,,m ∑ Probability of full system, Pm, is"Erlang's loss formula Exactly same expression for Pn of M/G/m system with Kem
M/M/m: finite system capacity, m; special case! …… 0 1 2 m-1 m l l l l l 3m m 2m (m-1)m mm for n m i n P m i i n n 0, 1, 2,... ! ( ) ! ( ) 0 = = å = m l m l • Probability of full system, Pm, is “Erlang’s loss formula” • Exactly same expression for Pn of M/G/m system with K=m