Little’s Law:
• Little’s Law overall station: • Little’s Law queue only:
E[N]: expected # customers& E[R]: response time & Labda: expected # customers arriving per unit time.
DTMC: Limiting distribution: (not always exist!) exist when: &
Recurrent states: return probability 1 (1-2 & 7 & 8-10)
Transient (nonrecurrent): non-zero chance not returning to this state (3-6)
Absorbing: probability state to self is 1 (7)
Period: greatest common divisor, if from state how many steps to get back.
State probability distribution after n steps:
CTMC: → q_i = how long in state, p_i = probability stransition from state
When in state i, probability going to state j within small amount of time h:
Generator matrix Q: count going out of state to other state, amount going in said state = negative (so total add up to 0)
Steady state: &
Period: no issue, reducibility is
GBE: look at states, flux in = flux out
MM1 queue: & || E[A] = exp interarrival times, E[S] expected service time
[Left CTMC i.e. constant rates]: GBE -> steady state dist → ^where p0:
→ & & normalization pi, we find:
Mean jobs E[N] then: & mean response time E[R] then:
Pr {R <= t}, , given erlang N distribution:
[Right CTMC, non-constant rates]: , then:
where p0:
Mean # jobs in system E[N]: , mean response time:
MMm queue: , Steady state dist: , Pr {waiting}: , w/ p0 = ↑
MM∞: , , then , , mean # jobs E[N] = rho
MM1m: ,
MMmm: , w/
MMmm queue w/ trunk reservation: K system users, issue job, think time Z, E[Z] = 1/labda. Job service time E[S]= 1/mu
Mean jobs:
MM1 q: . Where Lq is average # customers waiting in line: , with steady-state expected waiting time:
e.g. max size MM4/9 q: 9-4 = 5
MM3 q: 10 arrival per hour, w/ mean service time10 mins, traffic intens:
MD1 q: , , MG1: ,
Arrival + processing given, max arrival w_q <=3,
MM1 q: