Modélisation des files d'attente : Cours théorique

n
M/M/n0

M/M/1

S1
S2
S3
S4
S1S2S3S4
S1
S3
S2
S4
S1
S2
S3
S4

S1S2
S1↓S2
M/M/1
[0,∆t[,[∆t, 2∆t[, . . . , [(n−1)∆t, n∆t[, . . .
∆t
[(n−1)∆t, n∆t[
0 ∆t2∆t(n−1)∆t n∆t
Xn[(n−1)∆t, n∆t[
Xn
p∆t

E(Xn)≈p∆t
[(n−1)∆t, n∆t[p∆t=λ∆t
∆t→0+
P(Xn= 1) = λ∆t+o(∆t),
P(Xn= 0) = 1 −λ∆t+o(∆t),
P(Xn>2) = o(∆t).
t∈R+7−→ AtAt
[0, t]An∆t=X1+···+Xn
An∆tB(n, p∆t)t
t[(n−1)∆t, n∆t[
P(At=k)≈pk,∆t(t) = Ck
npk ∆t(1 −p∆t)n−k+o(∆t).
∆t0+n+∞n∆t∼t p∆t∼λt/n
pk,∆t(t) = n!
(n−k)! nk
(λt)k
k!1−λt nn−k
+o(∆t)
lim
∆t→0+pk,∆t(t)
P(At=k) = (λt)k
k!e−λt.
P(λt) (At)t∈R+
λE(At) = λt λ
λ=E(At)
t.
λ
T1= inf{t>0 : At>1}
T1> t t At= 0
P(T1> t) = P(At= 0) = e−λt
T1
FT1(t) = P(T16t) = 1 −e−λt.
T1E(λ)
fT1(t) = λ e−λt.
E(T1) = 1/λ λ
λ=1
E(T1).
T1Tn
(n−1) n
T1, T2, . . . , Tn, . . . E(λ)