(Joint work with Yoni Nazarathy, ex Swinburn Univ Tech)
(and Johan van Leeuwaarden, Eurandom and Eindhoven TU)
YN’s August 2010 seminar2011 QUESTA ‘problem’ papers by YN and DJD
Questions are of ‘Mathematical Interest’
(number-crunching construction of Year 12 aggregate)
(bibliometric measures: ‘impact factors’)
Balancing Reduces Asymptotic Variance of Outputs
Arrivals are Poisson at rate λ,Service times i.i.d. exponential at rate µ,
WHY the discontinuity at ρ := λ/µ = 1 ?
Is there similar behaviour with s servers ?
[NW08] includes a graph for systems M/M/s/(K −s)
‘correct’ family is for systems M/M/s/ s
‘Examine a system at its critical point(s)’
Branching process: (biological processes)Describe both sub- and super-critical behaviourTransition regime when mean offspring ≈ 1
In GI/GI/s, ρ = 1 is critical point.
Demarcation point between two stable phases.
ρ = 1: critical pt. when arrival rate λ, service rate µ/s,
For s-server system, have s servers each working at rate µ/s,
so system is ‘balanced’ when λ = s[µ/s] = µ. Write ρ = λ/µas in 1-server case, so ρ = 1 for ‘balance’.
Variance of Ndep has same asymptotics as for 1-server case
In M/M/s/K with ρ = 1, when s, K → ∞ in such a way that
for a function L(η) → 0 as η → ∞ [e.g. fix s at some finiteinteger ≥ 1).
When ρ = 1 − β/ s and s → ∞, there is non-trivial limit
behaviour but the limit f (η, β) say is no longer 2 .
(1) This is ‘QED’ regime: ‘Quality and Efficiency Driven’
(high utilization of servers with low probability of any appre-ciable waiting time) (Halfin & Whitt, c.1981)
(2) Find K, s → ∞ such that bothP (s) := Pr{arriving customer has no wait}
have positive limits . . . (solution: K = O( s ) ).
[ Re (Q.1): ρ = 1 −β/ s: limit discty vanishes at finer scale.]
We can find (expressions for) lims,K→∞ DM/M/s/K.
Let {πj} be the stationary distribution of the system-size pro-cess
Q(t): birth–death process on state space {0, 1, . . . , s + K} ={0, 1, . . . , J}.
We need formulae for second moments in terms of birth and
in M/M/1/K that comes from birth–death process expressiondue to Ward Whitt.
Limit relations (not given today) follow from (∗) via
gs(u) ps(du) for appropriate simple func-
tions gs, atomic measures ps and sets (intervals) As; look forweak cgce of measures and uniform cgce of functions.
(b) stationary probabilities πi are like Poisson probabilities —
use local CLT for individual terms or CLT for Poisson dis-tribution. EXCEPT: Need rate of convergence so use Berry–Esseen CLT to get next order term, or for local CLT, useFeller’s (1950/60/68) cgce of binomial probabilties to normaldensity yielding uniform bounds on error terms.
(c) (∗) is discrete sum over increasing number of terms — use
cgce of discrete sums to limit as for a Riemann integral.
converges to a process-limit (s → ∞).
How do we use this to give D (or appropriate analogue) ? ?[Recall: Ndep(·) consists of
So, approximation needs to be rectifiable (Brownian paths

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