Proof of Proposition 6

Recall Remark A.12.1. Recall also Lemmas 18 and 19, and their proofs. Note that the unobservable

pooled system is anM/M/N

system with the equilibrium effective arrival rate (A.219) and each unobserv-

able dedicated sub-system (that consists of one dedicated line and its server) is anM/M/1system with the

equilibrium effective arrival rate (A.222). Note also thatWf

p

(N x)represents the average sojourn time in the

represents the average sojourn time in theM/M/1system with the effective arrival ratexand service rate

µ.

We claim and show in Lemma 20 at the end of this section that

f

W

p

(N x)≤fW

d

(x)

forx < µ,

(A.223)

and hence

f

W

p

(N λ)≤Wf

d

(λ)

forρ <1.

(A.224)

Using (A.224), we will prove the claim in Proposition 6 under two main cases. Case 1: Suppose thatρ <1

and

_µ−1_λ

≤

R−_cf

. Then, by Lemma 19 and its proof,

cW

d

=

_µ−1_λ

,

qb

d

= 1and

bλ

e,d

=

λin equilibrium.

This and (A.224) imply thatWf

p

(N λ)≤

R−_cf

. Then, by the proof of Lemma 18,bq

p

= 1andbλ

e,p

=N λin

equilibrium. As a result,

c

W

p

=Wf

p

(N λ)≤Wf

d

(λ) =Wc

d

.

(A.225)

Recall the social welfare from (2.19), and recall thatbλ

e,d

N

=bλ

e,p

=N λ. Then,

d

SW

d

= (R−cWc

d

)λb

e,d

N

= (R−cWc

d

)λN

and

SWd

p

= (R−ccW

p

)bλ

e,p

= (R−ccW

p

)λN.

BecausecW

p

≤Wc

d

by (A.225),SWd

p

≥SWd

d

. This completes the proof of Proposition 6 under Case 1.

Case 2: Suppose now that either

ρ <

1

and

R−_cf

<

_µ−1_λ

, or

ρ

≥

1. From Lemma 19, it follows that, in

equilibrium, the average sojourn time in the unobservable dedicated system is

c

W

d

=

R−f

c

,

(A.226)

and the equilibrium social welfare in the dedicated system is

d

Given these performance metrics in the dedicated system, we now prove the claim by considering the fol-

lowing two subcases forWf

p

(N λ).

Case 2.1: Suppose thatfW

p

(N λ)

≥

R−_cf

. Then, by Lemma 18, the equilibrium average sojourn time in the

pooled system isWc

p

=

R−_cf

, which is equal tocW

d

by (A.226). Thus, by Lemma 18,

d

SW

p

= (R−cWc

p

)bλ

e,p

=fbλ

e,p

.

(A.228)

We now show that

b

λ

e,p

≥bλ

e,d

N.

(A.229)

Suppose for a contradiction thatλb

e,p

< Nbλ

e,d

. Then, (A.223) and the fact thatfW

p

(x)strictly increases in

xforx < N µimply that

c

W

p

=Wf

p

(bλ

e,p

)<Wf

p

(Nλb

e,d

)≤fW

d

(bλ

e,d

) =Wc

d

.

(A.230)

But, this contradicts withWc

p

=Wc

d

. Thus, we have (A.229). Based on (A.229), from (A.227) and (A.228),

it follows thatSWd

p

≥SWd

d

.

Case 2.2: Suppose thatfW

p

(N λ)<

R−_cf

. Then,qb

p

= 1andbλ

e,p

=N λin equilibrium. Thus,R−f−cWc

p

=

R−f

−cWf

p

(N λ)

>

0, which implies that the equilibrium long-run average sojourn time in the pooled

system satisfiesWc

p

<

R−_cf

=Wc

d

. Thus, the equilibrium social welfare in the pooled system is

d

SW

p

= (R−ccW

p

)λb

e,p

= (R−ccW

p

)N λ >(R−cWc

d

)N λ≥(R−cWc

d

)bλ

e,d

N

=SWd

d

,

which completes the proof of Case 2.2.

Note that combining Case 1 and Case 2 covers the entire parameter space. Thus, the claim in Proposition

6 follows.

The following lemma shows our claim in (A.223).

Lemma 20.

fW

p

(N x)≤Wf

d

(x)forx/µ <1.

Proof of Lemma 20:Suppose thatx/µ <1. Recall thatWf

p

(N x)represents the average sojourn time in the

M/M/N

system with the total effective arrival rateN xand the service rateµfor each server, andWf

d

(x)

represents the average sojourn time in theM/M/1system with the effective arrival ratexand service rate

µ. Denote byX

d

total number of customers inN

of theM/M/1lines in the steady-state, and letX

p

be the

corresponding figure in the aforementionedM/M/N

system. Based on this, to show our claim, we will use

standard likelihood comparison technique (see, for instance, (Smith and Whitt, 1981)). Letθ

p

(m+ 1)be the

transition rate from statem+1tomin the pooled system (in the steady-state),θ

d

(m+1|S

t

)be the transition

rate from statem+ 1tomin the dedicated system andS

t

is the state of the dedicated system (i.e., number

of customers in each of theN

lines) at timet, for anym= 0,1, . . .. Becauseθ

d

(m+ 1|S

t

)≤θ

p

(m+ 1)

for eachmregardless ofS

t

, in the steady-state, we have

P_(X

_d

_{=m)N x≤}P_(X

_d

_{=m+ 1)θ}

_p

_{(m+ 1)}

_and

P_(X

_p

_{=m)N x=}P_(X

_p

_{=m+ 1)θ}

_p

_{(m+ 1).}

Thus, we have

P_(X

_d

_{=m+ 1)}

P_(X

_d

_=m)

≥

N x

θ

p

(m+ 1)

=

P(X

p

=m+ 1)

P_(X

_p

_=m)

.

(A.231)

Using this, we now show thatE_(X

_d

_)≥E_(X

_p

_{). Note that (A.231) implies that}

P(Xd=j)

P_(Xd=i)

≥

P_(Xp=j) P_(Xp=i)

for

alli≤j,i, j∈N_{, which is equivalent to}

P_(X

_d

_=j)P_(X

_p

_=i)≥P_(X

_d

_=i)P_(X

_p

_=j).

_(A.232)

The summation on both sides of (A.232) overifrom0tojgives

P_(X

_d

_=j)P_(X

_p

_≤j)≥P_(X

_d

_≤j)P_(X

_p

_=j).

_(A.233)

Similarly, the summation on both sides of (A.232) overjfromi+ 1to∞results in

P_(X

_d

_{≥i+ 1)}P_(X

_p

_=i)≥P_(X

_d

_=i)P_(X

_p

_{≥i+ 1).}

_(A.234)

Combining (A.233) and (A.234) and lettingi=j=a, we have

P(X

d

≥a+ 1)

P_(X

_p

_{≥a+ 1)}

≥

P(X

d

=a)

P_(X

_p

_=a)

≥

P(X

d

≤a)

P_(X

_p

_≤a).

(A.235)

Thus,P_(X

_d

_≤a)≤P_(X

_p

_{≤a)for any non-negative integera, and hence}

E_(X

_d

_{) =}

∞

X

i=0

(1−P_(X

_d

_≤i))≥

∞

X

i=0

(1−P_(X

_p

_{≤i)) =}E_(X

_p

_).

_(A.236)

Observe that the long-run average number of customers in one of theN

separate dedicated sub-systems (i.e.,

e

L

d

) and the long-run average number of customers in the pooled system (i.e.,Le

p

) satisfy

Le

d

=E(X

d

)/N

andLe

p

=.

E(X

p

). Then, by Little’s Law and (A.236),

f

W

d

(x) =

e

L

d

(x)

x

=

E_(X

_d

_)/N

x

=

E_(X

_d

₎

N x

≥

E_(X

_p

₎

N x

=

e

L

p

(N x)

N x

=fW

p

(N x).

This completes the proof of the claim.

In document Tu_unc_0153D_19031.pdf (Page 135-139)