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??? ????M/M/n0 S 1 S 2 S 3 S 4 S 1 S 2 S 3 S 4 S 1 S 3 S 2 S 4 S 1 S 2 S 3 S 4 ??? ?S

1?????

??? ?S 2 S 1 S 2 ?????[(n-1)∆t,n∆t[????? ???? 0 ∆t

2∆t

(n-1)∆t n∆t P(Xn= 1) =λ∆t+o(∆t),

P(Xn= 0) = 1-λ∆t+o(∆t),

P(Xn>2) =o(∆t).

P(At=k)≈pk;∆t(t) =Cknpk∆t(1-p∆t)n-k+o(∆t).

??????? ??????∆t????0+?? ??????n????+∞????n∆t∼t?????? ????p∆t∼λt/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-t.

λ=E(At)

t

P(T1> t) =P(At= 0) =e-t

F

T1(t) =P(T16t) = 1-e-t.

f

T1(t) =λe-t.

λ=1

E(T1).

T

P(sn> t) =P(At< n) =n-1∑

k=0(λt)k k!e-t F sn(t) =P(sn6t) = 1-n-1∑ k=0(λt)k k!e-t. f sn(t) =F′s n(t) =λe-t-n-1∑ k=1[

λktk-1

(k-1)!-λk+1tk k!] e -t=-n-2∑ k=0λ k+1tk k!e-t+n-1∑ k=0λ k+1tk k!e-t. f sn(t) =λntn-1 (n-1)!e-t? ?? ??? ????T1,T2,...,Tn 0 s s s s n-1 s n s n+1 T 1 T 2 T 3 T n T n+1 P(Sn> s+t) =P(Sn> s)P(Sn> s+t|Sn> s) =P(Sn> s)P(Sn> t). ??? ??????? ?s? ′(s+t) =φ′(s)φ(t) ???? ????s= 0? ?? ??????φ′(0) =a? ′(t) =aφ(t).

P(Sn> t) =e-t

Q t=At-Dt W n+1= max(Wn+Sn-Tn+1,0) n=sn+Wn+Sn? t s t D t t Q t s S t s nsn+ 1σnW nSnW n+1Tn+1At D t ?? ??????Q∞=n????? ??? ??? ??????? ?????? ??? ????? ??? ????? ??????? ?? ??????Q∞=n???? ??? ?????Q∞=n-1??Q∞=n+ 1????? ???? n-1nn+ 1λλ ???? ??????? ?? ??????n? ???? ??????? ?? ?????? ??????? ?n-1-→n←-n+ 1? ?? ???? ??????? ?? ??????n??? ????λπn-1+µπn+1? ?? ?????? ??? ????? ???? ?? ??????? ?n-1←-n-→n+ 1? ?? ???? ??????? ?? ??????n??? ????(λ+µ)πn? {λπ0=µπ1 n-1+µπn+1= (λ+µ)πn??n>1. µr

2-(λ+µ)r+λ= 0

n=αρn+β. n=0π n= 1? ?? ??? ??????? ?? n=0ρ n=1 n=P(Q∞=n) = (1-ρ)ρn, n∈N.

E(Q∞) =λ

t s t D t

P(W∞6t) =∞∑

n=0P(W∞6t|Q∞=n)P(Q∞=n).

P(W∞6t) =π0+∞∑

n=1π n∫ t 0µ nsn-1 (n-1)!e-sds = (1-ρ)[

1 +ρµ∫

t

0∞

n=0(ρµs)n n!e-sds] = (1-ρ)[

1 +λ∫

t 0 e-(1-)sds]

P(W∞6t) = 1-λ

e-(-)t.

E(W∞) =∞∑

n=0E(W∞|Q∞=n)P(Q∞=n).

E(W∞) =∞∑

n=0n

P(Q∞=n) =1

E(Q∞),

E(W∞) =λ

E(µ-λ)?

P(W∞+S∞6t) = 1-e-(-)t.

E(W∞+S∞) =1

E(Q∞) =µE(W∞) =λE(W∞+S∞). λ??µ?? ????? ??? ?? ????? ?? ??????W∞?????? ?? ??????? ??????µ-λ?

1=ρ2=2

3

E(W1;∞) = 20??,E(W2;∞) = 10??,

E(W1;∞+S1;∞) = 30??,E(W2;∞+S2;∞) = 15??. tQ t B ∞I∞ f 1 t ??I1(z) =+∞∑ n=01 (n+ 1)!n!( z 2 0 tfB∞(t)dt? ?? ??? ????? ???

E(B∞) =1

E(B∞) =E(Q∞|Q∞>1)×E(S∞).

E(Q∞|Q∞>1) =E(Q∞)

P(Q∞>1)=µ

µ-λ??E(S∞) =1

??????? ??????? ?? ??? ??? ????? ???? ?? ???E(λ)? f

I∞(t) =λet.

?????τn????? ???? ????? ???Sn? I ?? ??????? ?? ?????Sn? ?? ? ???? ?? ???τn=In-1+Sn? n={Sn??Qn-1>1? S n+In-1??Qn-1= 0? t s n-1snσn-1σn nS nAt D t Q n-1>1 t s n-1snσn-1σnIn-1Sn nS nAt D t Q n-1= 0 P(τn6t) =P(Qn-1>1)P(Sn6t|Qn-1>1) +P(Qn-1= 0)P(In+Sn6t|Qn-1= 0). f

∞(t) =P(Q∞>1)fS∞(t) +P(Q∞= 0)fI∞+S∞(t) =ρµe-t+ (1-ρ)(fI∞⋆ fS∞)(t)

(fI∞⋆ fS∞)(t) =∫ t 0 t 0 e-se-(t-s)ds=λµ

µ-λ(e-t-e-t).

f ∞(t) =λe-t. S 1 S 2 S 1 S 2 S n f

S(t) =n∑

k=1α 16j6n j̸=k(

1-µk

j)

E(S) =n∑

k=11 k.

E(n;µ)?

P(N=k) =pk,16k6n.

S 1 S 2 S n p 1 p 2 p n

P(SN6t) =n∑

k=1P(N=k)P(SN6t|N=k) n∑ k=1P(N=k)P(Sk6t) n∑ k=1p k(1-e-ktk). f

SN(t) =n∑

k=1p kµke-kt.

E(SN) =n∑

k=1p k k. k=1p k= 1? f

SN(t) =n∑

k=1p kµe-t=µe-t ???????B?????

P(A|B) =P(A∩B)

P(B).

P(A) =n∑

i=1P(Bi)P(A|Bi).

E(X) =+∞∑

n=0nP(X=n)

E(X) =∫

0 tfX(t)dt? var(X) =E[(X-E(X))2] =E(X2)-[E(X)]2? ????N?? ?????? ???

P(X+Y=n) =+∞∑

k=0P(X=k)P(Y=n-k) f

X+Y(t) = (fX⋆ fY)(t) =∫

t 0 fX(s)fY(t-s)ds. ??? ?? ???????P(λ)?

P(X=n) =λn

n!e-????n∈N?E(X) =λ?var(X) =λ? f

X(t) =µe-t??FX(t) = 1-e-t????t∈R+?E(X) =1

?var(X) =1 2?

P(X=n) =ρ(1-ρ)n-1????n∈N∗?E(X) =1

?var(X) =1-ρ 2? f

X(t) =λntn-1

(n-1)!e-t????t∈R+?E(X) =n ?var(X) =n 2? 16j6n j̸=k(

1-λk

j) f

X(t) =n∑

k=1α kλke-kt????t∈R+?E(X) =n∑ k=1α k k?var(X) = 2n∑ k=1α k 2k-( n∑ k=1α k k) 2 k=1p k= 1? f

X(t) =n∑

k=1p kλke-kt????t∈R+?E(X) =n∑ k=1p k k?var(X) = 2n∑ k=1p k 2k-( n∑ k=1p k k) 2

P(Q∞=n) =

0ρn

n!??06n < n0 0nn00 n 0!( n 0) n0??n>n0 ????π0=P(Q∞= 0) = n

0-1∑

j=0ρ j j!+ρn0 n 0!(1- n

0)

-1

E(Q∞) =ρ+π0ρn0+1

(n0-1)!(n0-ρ)2,

P(W∞6t) = 1-ρn0

n 0!(1- n

E(W∞) =π0ρn0

n 0!(1- n 0)1 n

0µ-λ.

E(Q∞) =λE(W∞+S∞).

P(Q(t) =m) =+∞∑

n=0P(A1(t) =m+n)P(A2(t) =n).

P(Q(t) =m) =e-(1+2)t+∞∑

n=0(λ1t)m+n (m+n)!(λ2t)n n!=(λ1 2) m=2 e -(1+2)t+∞∑

1λ2)m+2n

(m+n)!n!

P(Q(t) =m) =(λ1

2) m=2 e

1λ2)

??Im(z) =+∞∑ n=01 (m+n)!n!( z 2

1λ2)??? ??

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