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Asymptotically Optimal Control of Many-server Heterogeneous
Service Systems with H2
Service Times
Tolga Tezcan
Simon School of Business
University of Rochester
Rochester, NY
[email protected]
June 16, 2011
Abstract
Optimal control of many-server heterogenous service systems with
service times that havea special hyper-exponential distribution,
denoted by H
2, which is a mixture of an exponential
distribution and a unit point mass at 0, is considered. A static
priority policy that assignspriorities to server pools based on
their service time distributions is proposed. This policy isshown
to be asymptotically optimal in the many-server heavy traffic
regime in minimizing thetotal number of customers in the system or
in the queue under two different assumptions onservice time
distributions.
Keywords: Heterogeneous servers; Many servers; Heavy traffic;
Halfin-Whitt regime; Asymptoticoptimality
1 Introduction
The heavy-traffic analysis approach has been successfully used
to analyze complex queueing systemsthat cannot be analyzed using
standard queueing techniques for exact analysis; e.g., see [21, 8,
4].In this approach, the queue lengths in a heavily loaded system
are shown to be close to a diffusionprocess under reasonable
scheduling policies. The many-server heavy-traffic analysis,
initiated bythe seminal paper of [20], has been used to analyze
queueing systems with many servers in heavy-traffic. Unlike the
conventional heavy-traffic analysis, the many-server analysis is
more suitable forsystems with several servers working in parallel,
see [16] for a comparison of these two regimes.In this paper we
establish asymptotically optimal policies for a service system with
heterogenous
servers in a many-server heavy traffic regime. We consider
service times with a special hyper-exponential distribution in
order to gain some insights for the optimal control of
many-serversystems under non-exponential service times.
The heterogenous service systems we study consist of a single
customer class and multiple serverpools. In each server pool there
are several servers and two servers that belong to the same
pool
Research supported by NSF Grant CMMI-0954126.
1
nuscript
k here to download Manuscript: rev4_11_06_10.pdf Click here to
view linked Refe
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are assumed to have the same service time distribution. These
systems are known as inverted-Vsystems; see [16] and [1]. Our main
goal is to devise control policies that minimize the congestionin
these systems. The control of inverted-V systems consists of two
parts. First, if an arrivingcustomer finds idle servers in
different pools, the control policy must specify which server pool
thearriving customer will be routed to. It is also possible to hold
the customer in queue even though
there are idle servers. Second, when a server finishes serving a
customer and there are customerswaiting in the queue at that time
point, the control policy must specify whether the server
shouldstart serving another customer or it should idle.
We assume that the service times have H2 distribution, that is,
a customer that is routed toserver j has an exponential service
time distribution with rate j with probability pj or
customersservice time is equal to zero. Although, having service
times equal to zero is not possible in most ofthe real systems, it
may be used to approximate very short service times. In addition it
provides uswith additional insights on the control of inverted-V
systems when service times are not exponential.
Before we explain our proposed policy we first introduce our
assumption on service time distri-butions.
Assumption 1. One of the following conditions hold.
i. Xj st X1, for all j = 2, . . . , J ,ii. 1 j, for all j = 2, .
. . , J ,
where Xj is a random variable which has the same distribution
with the service times in pool j.
Although they look similar, these two conditions are different.
Because
P(Xj > t) = pj exp{jt}, (1.1)
it is easy to verify that Assumption 1(i) holds if and only ifj
1 and pj p1, for all j = 2, . . . J .Therefore, the first
assumption implies the second one but the second one does not imply
the first
one in general. We propose the following static priority
policy
At the time of a customer arrival, route the arriving customer
to one of the server pools withj 2 with available servers, if there
is an available server in those pools. For asymptoticanalysis it is
immaterial which server pool is chosen but for concreteness we
choose the highestindexed server pool. Otherwise, if there is an
available server in pool 1, route the customerto server pool 1. If
all the servers are busy, the arriving customer joins the queue and
startswaiting.
At the time of a service completion, the server finishing
service picks the longest waitingcustomer in the queue (so our
policy is non-idling). If there are no customers waiting theserver
idles after finishing service of a customer.
We denote this policy by . The main result of this paper is
that, under Assumption 1, isasymptotically optimal as the arrival
rate and the number of servers get large and the load on thesystem
approaches its capacity at a certain rate (see Section 2 for a
detailed definition). Specifically,we show that asymptotically
minimizes the average number of customers in the system
underAssumption 1(i) and in the queue under Assumption 1(ii) over
any finite time interval in the senseof stochastic ordering or
expected value in the limit.
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In [1], inverted-V systems with exponential service times have
been analyzed in a similar asymp-totic regime as we consider. It is
shown that the fastest-server-first (FSF) policy, which routes
anarriving customer to one of the available servers with the
smallest expected service time, is asymp-totically optimal in
minimizing the number of customers in the system over finite time
intervalsand in steady state. In our model, if the service times
are exponential, i.e., pj = 1 for each pool
j, the proposed policy
reduces to the fastest-server-first (FSF) policy (if we assume
in additionthat servers with lower mean service time are given
priority. For asymptotic optimality this is notnecessary). Note
that if service times are exponential, Assumption 1(ii) and
Assumption 1(i) areequivalent. Optimality under Assumption 1(i) is
very intuitive since service times are assumedto be stochastically
ordered. On the other hand, only the distribution of the non-zero
part of theservice times matters in Assumption 1(ii) when pj <
1. It is possible to construct examples where aserver pool having
priority over another one may have service times with higher mean
and variance(see Section 4.2 for details).
Optimal control of queueing systems with non-exponential service
times in the many-serverheavy traffic regime has not been treated
in the literature to the best of our knowledge (paper [28],which
appeared after the first version of this paper, considers the
control of V-model systems usingfluid models). We take a first step
to treat this problem. Although the domain of the systems
we consider may seem to be too limited, the results and ideas in
[1] for inverted-V-systems withexponential service times have
served as a stepping stone for understanding the control of
complexsystems with many servers; see for example [18] and [7].
The general idea of our proof of asymptotic optimality has now
become standard in heavy trafficanalysis; see [8, 9, 4, 13, 12]. We
first establish an asymptotic lower bound for the total numberof
customers in the system (or the queue length) and then show that
the relevant process under achieves the lower bound in the limit.
When the service times have exponential distributionit is common in
the literature to first determine an asymptotically optimal
preemptive policyand then show that the values of the objective
function under a similar non-preemptive optimalpolicy converges to
the same limit; see [6, 1]. Such an approach is not possible when
the servicetimes have hyper-exponential distributions; see Section
4.3 for more details. Hence, to study the
optimal control of inverted-V models, we define a mapping with
an appropriate domain and rangeand show that the the total number
of customers in the system (or the queue length) under thismapping
is minimized among all feasible mappings. We only consider H2
service times becausea direct extension of our optimality proof
does not seem possible to more general distributionssuch as
phase-type. However, we believe that our results can be extended to
the case with generalhyper-exponential service times under
Assumption 1(i) or under a slightly different version ofAssumption
1(ii).
We close this section with a review of the related literature.
The many-server asymptoticregime we focus on is first proposed by
[20]. The optimal control of parallel-server systems inthe
many-server regime have been studied in [2, 3, 1, 13, 12, 18, 19,
17, 6, 5, 7]. The maindifference between our work and the existing
literature is that we do not restrict our attention
to exponential service times. Although our research is the first
to address the optimal control insystems with non-exponential
service times, diffusion and fluid limits of systems with general
servicetimes distributions have been established; see [25] for the
treatment of phase-type distributions andsee [26, 22] for general
distributions. Also, [28] considers the control of V-model systems
in many-server regime with general service times using a fluid
scaling.
The rest of this paper is organized as follows. In Section 2 we
present the details of the
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queueing model and the asymptotic regime we study. We define a
regulator mapping and establishits properties in Section 3. We
present our main results in Section 4. In addition, we present
thedetails of simulation results and discuss the differences
between preemptive and non-preemptivescheduling in our setting. The
remaining sections are devoted to the proof of our main results.
Inthe next section we collect the notation and terminology used in
the rest of this paper.
1.1 Notation
Let |x| denote the max norm on Rd given by |x| =
maxi=1,2...,d{|xi|}. For x R, x = (x) 0and x+ = x 0. For each
positive integer d, Dd[0, ) denotes the d-dimensional Skorohod
pathspace; see [14]. For x, y Dd[0, ) and 0 s < T we set
x() y()[s,T] = supstT
|x(t) y(t)|.
and we write x() y()T = x() y()[0,T] for notational convenience.
We also setx() y()(s,T] = sup
s 0 and f(0) = f(0) by convention.The space Dd[0, ) is endowed
with the Skorohod J1 topology and the weak convergence in
this space is considered with respect to this topology. For a
sequence of functions {xr} in Dd[0, ),the sequence is said to
converge uniformly on compact sets to x Dd[0, ) as r , denoted byxr
x u.o.c., if for each T > 0
xr() x()T 0 as r .
2 The queueing model and asymptotic framework
In this section we first describe the details of the queueing
model and the asymptotic framework
we consider. Then we present the queueing equations.We consider
an inverted-V system with J 2 server pools (we set J= {1, . . . , J
}). We assume
that servers in the same pool have the same service time
distribution. The service times are assumedto have a special
distribution, denoted by H2 as in [30]. To recap, the service times
in pool j aremixtures of an exponential distribution with rate j
with probability 0 < pj 1 and a unit pointmass at 0 with
probability 1 pj.
In our asymptotic analysis, we consider a sequence of inverted-
V-systems indexed by r withthe same structure. The arrival rate and
the number of servers go to infinity in this sequence; inthe rth
system the arrival rate is given by
r = r (2.1)
and the number of servers, Nrj , in the jth pool is given by
Nrj = jr, for j J, (2.2)for some j > 0. (One might require
N
rj to be an integer by rounding and this does not affect our
analysis. For notational simplicity we do not use rounding.) Let
mj(= pj/j) denote the averageservice time in pool j.
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Let Qr(t) denote the number of customers in the queue, Zrj (t)
denote the number of customersbeing served in pool j and Yr(t)
denote the total number of customers in the system at time t inthe
rth system. The diffusion scaled processes are defined by
Qr(t) =Qr(t)
rand Zrj (t) =
Zrj (t) Nrjr
, for j J, (2.3)
and
Yr(t) = Qr(t) +J
j=1
Zrj (t).
Obviously these quantities depend on the policy, , used. When we
want to make this dependenceexplicit we append to these quantities
as a subscript.
The arrival process in the rth system Ar() is defined by
Ar(t) = sup{m 0 :m=0
u() rt}, (2.4)
where {u(l) : l = 1, 2, . . .} is a sequence of i.i.d.
nonnegative random variables with mean 1 andvariance 2 [0, ) and
u(0) is an arbitrary nonnegative random variable. By convention,
emptysums are set to be zero.
As described in the introduction, control policies are needed to
determine how inverted-V modelsoperate. In search of an optimal
control policy, we restrict our attention to admissible
controlpolicies that are non-idling, head-of-line, non-preemptive
and Markovian as described in [13].
We make the following (so called) many-server heavy traffic
assumption; for R,J
j=1
Nrjmj
= r +
r, for r 2. (2.5)
We also assume that the following holds for the sequence of
initial conditions;
(Qr(0), Zr(0)) (Q(0), Z(0)), as r , (2.6)
where denotes weak convergence, (Q(0), Z(0)) is a random vector
and Zr =
Zrj , j J
.
Queueing processes: Next we provide the details of the queueing
processes. Let Sj denote aPoisson process with rate j, j J, and
assume that Sjs, j J, are independent. We use Arj(t),j J, to denote
the number of customers whose service started in server pool j
before time t. Wenote that Arj(t) is nondecreasing for all j J
because we only consider non-preemptive policies.Let Drj (t), j J,
denote the number of service completions by time t in server pool j
whose servicetimes were not equal to 0. We can write
Drj (t) = Sj
t0
Zrj (s)ds
, for j J,
for any admissible policy, see Theorem 2.1 in [13] for details.
We set
Ar =
Ar, Arj : j J
, Zr =
Zrj : j J
, and Dr =
Drj : j J
.
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Let Xr = (Ar, Zr, Qr, Dr). Note that Xr depends on the control
policy, , used. When we want tomake this dependence explicit we
append to the process Xr or its components.
Let {j(n), n 1} be a sequence binomial i.i.d. random variables
such that j(1) is equal to 1with probability pj and equal to zero
with probability 1 pj. We can write the evolution of thequeuing
processes as follows;
Qr(t) = Qr(0) + Ar(t) Jj=1
Arj(t), (2.7)
Zrj (t) = Zrj (0) +
Arj(t)
n=0
j(n) Drj (t) for j J. (2.8)
Next, we write the queueing equations (2.7)-(2.8) in a form more
amenable to analysis. Thefluid scaling is defined by
Xr = Xr/r.
We define the diffusion scaled departure processes by
Drj (t) =
r
Sj t0 Zrj (s)dsr
j
t
0Zrj (s)ds
, j J (2.9)and the diffusion scaled arrival process by
Ar(t) = r1/2 (Ar(t) rt) .Let
Crj (t) =
t=1
j() and Crj (t) = r
1/2
Crj (rt) rtpj
, j J.
We set Cr = Cr1 , . . . , CrJ and definearj(t) = C
rj (A
rj(t)), j J. (2.10)
Also let
urj(t) = r1/2
Arj(t)
Nrjmj
t
, j J (2.11)
and ur =
urj ;j J
. The process urj can be interpreted as the diffusion scaled
deviation of
number of customers routed to queue j from its nominal value.
For notational convenience weset
wrj (t) = Zrj (0) + a
rj(t)
Drj (t), j
Jand wrq(t) = Q
r(0) + r1/2 (Ar(t)
rt) + t. (2.12)
By (2.7)(2.12),
Zrj (t) = wrj (t) j
t0
Zrj (s)ds +pjurj(t), j J, (2.13)
Qr(t) = wrq(t) jJ
urj(t), (2.14)
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and
0 Qr(t) and Zrj (t) 0 for j J. (2.15)
The non-idling condition implies that
Qr(t)J
j=1
Zrj (t) = 0, for all t 0. (2.16)
Depending on the policy used additional equations are added to
(2.13)-(2.15). Under the pro-posed policy the following condition
also holds;
Jj=j
Arj(t) Arj(s)
= Ar(t) Ar(s), (2.17)
if
Jj=j Z
rj (u) < 0 for all u [s, t] and for j J.
3 Mapping
In this section we define the mapping that we later use to
characterize the optimal limitingqueueing processes and to study
the limit of queuing processes under the proposed policy . Letx =
(xq, xj : j J) DJ+1[0, ), u = (uj : j J) DJ[0, ), z = (zj : j J)
DJ[0, ) andq D[0, ). Consider the following equations
zj(t) = xj(t) jt0
zj(s)ds +pjuj(t), for all j J, (3.1)
q(t) = xq(t)
J
j=1
uj(t), (3.2)
zj(t) 0, for all j J, (3.3)q(t) 0, (3.4)
q(t)
Jj=1
zj(t) = 0, (3.5)
for t 0. We highlight the fact that equations (2.13)-(2.16) are
very similar to (3.1)(3.5). Theprocesses Arj s, hence u
rj s, see (2.11), determine how a queueing system is controlled.
In this setting
the control is determined by u. We start with defining feasible
controls in this context.
Definition 1. Given x DJ+1
[0, ), u DJ
[0, T] is said to be a feasible control if there existz DJ[0, )
and q D[0, ) that satisfy (3.1)(3.5).Given x and a feasible control
u, the process (z, q) satisfying (3.1)(3.5) is unique; see
Theo-
rem 4.1 in [24]. In order to make this dependence explicit we
write ( zu, qu) to denote these processesassociated with a feasible
control u when the control is not explicit from the context.
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Next we define the mapping . Let : DJ+1[0, ) DJ[0, )D[0, )DJ[0,
) be definedfor x DJ+1[0, ) by
(x) = (1(x), 2(x), 3(x)) = (z, q, u),
where (z, q, u) satisfies (3.1)(3.5) and the following
condition
zj(t) = 0, for j 2, and t 0. (3.6)
Note that by (3.1) this implies
uj(t) = xj(t)/pj, for j 2 and t 0. (3.7)
In words, mapping makes sure that all the servers in pools j =
2, . . . , J (those that havepriority over the first pool) are busy
at all times, see (3.6). Obviously depends on p but we donot
explicitly indicate this dependence in our notation because we
assume that p is fixed throughoutthe paper. Next we show that is
well defined.
Lemma 2. For each x DJ+1
[0, ), there exists a unique (z, q, u) that satisfies
(3.1)(3.6).Therefore, the mapping is well defined. In addition, is
continuous provided that the functionspaces DJ+1[0, ) and DJ[0, )
D[0, ) DJ[0, ) are endowed with either the topology ofuniform
convergence over compact intervals or with the Skorohod-J1
topology.
Proof. Let x DJ+1[0, ). Clearly, uj for j 2 is well defined by
(3.7). Let : DJ+1[0, ) D[0, ) be defined by (x) = y1, where
y1(t) = y1(0) + xq(t) +
Jj=1
xj(t)
pj+ 1
t0
(y1(s))ds.
By Theorem 4.1 in [24], is well defined, i.e., there exits a
unique y1D[0,
), and it is
continuous provided that the function spaces DJ+1[0, ) and D[0,
) are endowed with either thetopology of uniform convergence over
compact intervals or the Skorohod J1 topology. Let
q(t) = (y1(t))+ and z1(t) = p1(y1(t)). (3.8)
Hence,
y1(t) = q(t) +z1(t)
p1. (3.9)
Clearly q and z1 satisfy (3.3)(3.5). Define
u1
(t) =z1(t) x1(t) + 1
t0 z1(s)ds
p1.
Note that z1 and u1 satisfy (3.2). Also q satisfies (3.2) by
(3.7) and (3.9). This proves existence.Continuity and measurability
follows from the continuity of and (3.8).
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Now we prove that the mapping is optimal in a certain sense.
Given a feasible control u andassociated processes q, z satisfying
(3.1)(3.5) with u, let
y(t) = q(t) +J
j=1zj(t).
We write yu instead of y when we want to make the dependence on
the control u explicit.
Theorem 3. Fix x DJ+1[0, ) and let u = 3(x). Then for any other
feasible control u andfixedT > 0,i. if Assumption 1(i) holds
then
yu(t) yu(t), t [0, T], (3.10)
ii. if Assumption 1(ii) holds then
(yu(t))+ (yu(t))+, t [0, T]. (3.11)
Proof. Fix x DJ+1[0, ). Let u be a feasible control and denote
by (z, q) associated processessatisfying (3.1)(3.5). First we prove
(ii). Define
yu(t) = q(t) +J
j=1
zj(t)
pj.
By (3.3)(3.5)
(yu(t))+ = q(t) and (yu(t))
= J
j=1
zj(t)
pj. (3.12)
We also have by (3.1) and (3.2)
yu(t) = yu(0) + xq(t) +J
j=1
xj(t)
pj
Jj=1
j
t0
zj(s)
pjds.
We use Lemma 4.4 in [12] to complete the proof. We define
r(t) =
t0
q(s)ds and vj(t) =
t0
zj(s)
pjds. (3.13)
We also set
w(t) = y(0) + xq(t) +
J
j=1
xj(t)
pj.
(This quantity is not related to wrq defined in (2.12).) We
have
yu(t) = w(t) J
j=1
jvj(t).
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By (3.12) and (3.13), y, r and vj, j J, satisfy the conditions
of Lemma 4.4 in [12], henceyu(t) yu(t), t [0, T]. (3.14)
This inequality gives (3.11) by (3.5) and (3.12).We prove (3.10)
using (3.14), Assumption 1(i) and (3.11). Note that Assumption 1(i)
implies
by (1.1) thatp1 pj, for all j = 2, . . . , J . (3.15)
Therefore,
(yu(t)) (a)=J
j=1
zj(t)(b)
p1J
j=1
1
pjzj(t)
(c)= p1 (yu(t))
(d)
p1 (yu(t)) (e)= (yu(t)) , (3.16)
where (a) follows from (3.3) and (3.4), (b) follows from (3.3)
and (3.15), (c) follows from (3.12), (d)follows from (3.14) and (e)
follows from (3.6). Because Assumption 1(i) implies Assumption
1(ii),(3.10) still holds (see the discussion after Assumption 1).
Hence, combined with (3.11), (3.16) gives(3.10).
4 Main results and Insights
In this section, we first present our main results. Then we
present the results of simulation experi-ments in Section 4.2 and
comment on our proof technique in Section 4.3.
4.1 Main Results
Let B = (Wq, W), where Wq and W = (Wj : j J) are independent
Brownian motions with drifts and 0, variances 2 and (jj(2 pj) : j
J), respectively, and initial states
Wq(0) = Q(0) and Wj(0) = Zj(0) for j
J.
Let
(Z, Q, u) = (B)
and
Y(t) = Q(t) +J
j=1
Zj (t).
First we prove that Q and Y provide a lower bound for all the
admissible policies.
Theorem 4. Consider a sequence of inverted-V systems. Assume
that (2.1), (2.2), (2.5) and (2.6)hold. For any sequence of
admissible policies {r} and for any T > 0,i. under Assumption
1(i)
lim infr
P
T0
Yrr(t)dt > x
P
T0
Y(t)dt > x
(4.1)
for any x > 0 and
liminfr
E
T0
Yrr(t)dt
E
T0
Y(t)dt
, (4.2)
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Setting 1 p1 2 (E[X1] ,E[X2]) (Var[X1] ,Var[X2])
1 3 0.7 4 (0.233,0.25) (0.101,0.0625)2 3 0.5 4 (0.166,0.25)
(0.083,0.0625)3 3 0.3 4 (0.1,0.25) (0.056,0.0625)
Table 1: Service time parameters
ii. under Assumption 1(ii)
lim infr
P
T0
Qrr(t)dt > x
P
T0
Q(t)dt > x
(4.3)
for any x > 0 and
liminfr
E
T0
Qrr(t)dt
E
T0
Q(t)dt
. (4.4)
Next we show that the cost under the proposed policy coincides
with the lower bound in thelimit.
Theorem 5. Consider a sequence of inverted-V systems. Assume
that (2.1), (2.2), (2.5) and (2.6)hold. Under policy , for any T
> 0,i. under Assumption 1(i)
limr
P
T0
Yr(t)dt > x
= P
T0
Y(t)dt > x
(4.5)
for any x > 0 and
limr
ET
0Yr(t)dt = E
T
0Y(t)dt , (4.6)
ii. under Assumption 1(ii)
limr
P
T0
Qr(t)dt > x
= P
T0
Q(t)dt > x
(4.7)
for any x > 0 and
limr
E
T0
Qr(t)dt
= E
T0
Q(t)dt
. (4.8)
Remark 6. The main difference between parts (i) and (ii) in
Theorems 4 and 5 is that underAssumption 1(i) we prove a slightly
stronger result; the total number of customer in the system is
minimized (in the appropriate sense). Whereas only the number of
customers in queue is shown tobe minimized under the proposed
policy under Assumption 1(ii).
4.2 Simulation experiments
In this section we focus on inverted-V model systems with J = 2
and carry out some simulationexperiments. (The reader who wants to
see the details of the proofs first can skip this section
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Exp No N 1 p1 2 Q
1 (50,50) 407 3 0.7 4 1.70 (1.31)2 (50,50) 480 3 0.5 4 0.63
(0.03)3 (50,50) 685 3 0.3 4 5.87 (5.31)4 (200,200) 1643 3 0.7 4
3.35 (2.08)5 (200,200) 1985 3 0.5 4 4.19 (
3.27)
6 (200,200) 2770 3 0.3 4 5.38 (3.97)
Table 2: Simulation results
and go to Section 5.) We set J = 2, p2 = 1, 1 = 3, and 2 = 4.
Therefore, Assumption 1(ii)holds (irrespective of the value of p1
(0, 1]) and the policy that gives priority to the second poolshould
be asymptotically optimal. In our experiments, we compare the
proposed policy (whichgives priority to the second pool) to the
policy that gives priority to the first pool. We considerthree
different settings for service times by altering p1 as presented in
Table 1. We also presentthe expected service times, under column
(E[X1], E[X2]) and the variances of the service times,
under column (Var[X1],Var[X2]) for pools 1 and 2, respectively,
under three different values for p1in Table 1. In all these
settings, the expected service time for the first pool is smaller
than thatof the second pool. The variance of the service times for
pool 1 is higher than that for the secondpool in the first two
settings and lower in the last experiment.
The details and results of the simulation experiments are
presented in Table 2. Initially weconsider mid-size systems in
Experiments 1 through 3 in Table 2 with 50 servers in each pool
andwith the service time distributions given as in each setting in
Table 1. We run three additionalexperiments, Experiments 4 through
6 in Table 2, using similar parameters to Experiments 1through 3,
respectively, except that the number of servers in each pool in
these experiments isequal to 200. The column N is the number of
servers in each pool and is the arrival rate. Thelast column, Q, is
the difference between the average queue lengths throughout the
simulation
when we give priority to the first and the second pools,
respectively. In parentheses, we displaythe 95% confidence interval
we obtain from 100 replications. In each replication, we simulate
thesystem to allow 1,000,000 arrivals and we start each simulation
with an empty system.
The results of our experiments show that the proposed policy
performs better than the policythat gives priority to the first
pool and the differences between queue lengths are
statisticallysignificant in all the experiments. We emphasize the
fact that in Experiments 3 and 6, both themean and variance of
service times in the second pool are larger than those in the first
pool, but itis still optimal to give priority to the second
pool.
4.3 Preemptive vs. nonpreemptive policies
We comment on our proof technique before we provide the details
of the proofs. In the many-server
asymptotic analysis, when service times have exponential
distribution, a common methodology toshow that a non-preemptive
policy is asymptotically optimal is to first show that a similar
butpreemptive policy is optimal and then to show that the
performance of the non-preemptive policycoincides with that of the
optimal preemptive policy in the limit. For example, in [ 1] the
followingpreemptive policy is analyzed first; if a faster server
idles, the service of a customer in the slowerserver pool (if there
is any) is preempted and that customer is re-routed to the faster
pool. Then
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it is shown that the policy that does not allow preemptions have
the same asymptotic performanceas this preemptive policy.
However, such an approach cannot be used in the current context.
To illustrate, consider aninverted-V model with two server pools.
If preemptive policies are allowed, it can be shown that thenumber
of customers in queue converges to zero in the many-server heavy
traffic regime for certain
systems as we explain below. Assume that we reserve one tagged
server in the first pool and thattagged server is used differently
from other servers in a way we explain next. Upon a new
customerarrival, we send the arriving customer to the tagged
server. Note that with probability 1 p1 thenew customers service
time is zero and if so that customer will leave the system
immediately. Ifthat customers service time is not zero, we preempt
this customers service. If possible we try toassign this customer
to a server in pool 2 and then to a server (other than the tagged
server) inpool 1. Otherwise, the customer starts waiting in the
queue until a server other than the taggedserver becomes available.
(Although the policy we describe is idling, a non-idling version
can beconstructed similarly.)
Consider the following parameters, 1 = 2 = 1, p1 = 0.5 and p2 =
1, and Nr1 = N
r2 = 50.
Therefore, under the preemptive policy described above 50% of
the customers will leave the systemimmediately upon arrival. The
total system capacity is 150 customers per unit time by (2.5).
However if preemption as described above is allowed, the system
can handle 200 customers perunit time (50% of them will leave the
system immediately and the rest will have average servicetime equal
to 1). Therefore, even when (2.5) holds, the system is not under
heavy traffic whenpreemptions are allowed. Using this fact, the
diffusion scaled queue length can be shown to convergeto zero at
all times for large enough r when (2.1), (2.2), (2.5) and (2.6)
hold. This is not possibleunder any non-preemptive policy as shown
in our main result. Therefore, we work directly withthe mapping
defined in Section 3 instead of considering preemptive policies in
our proof.
5 Asymptotic Bounds for admissible policies
The rest of the paper is devoted to the proofs of Theorems 4 and
5. In this section we providebounds for Qr, Zr and ur under
non-idling and non-preemptive policies. These bounds will beused in
several places throughout the proofs of our main results. In
Sections 6 and 7 we proveTheorems 4 and 5, respectively.
For a fixed admissible policy , let
wr =
wrj , j J
, br =
wrq , wr
, (5.1)
(recall (2.12)) and
(Zr, Qr, ur) = (br), (5.2)
where the mapping is defined as in Section 3. Note that the
process br depends on the policy but we ignore this dependence from
our notation for simplicity.
Lemma 7. For any admissible policy and T > 0
Qr()T Zr()T ur()T CTbr()
T+Ar()
T
,
for some CT > 0, independent of the policy and for all j
J.
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The following result can be proved similarly to Lemma 7 hence
the proof is omitted.
Lemma 8. Fix an admissible policy and let (Zr, Qr, ur) be
defined as in (5.2). For T > 0
Qr()T Zr()T urT CT
br()T
+
Ar()T
,
for some CT > 0, independent of the policy.
Proof of Lemma 7. Fix an admissible policy and T > 0. First
note that by (2.15), (2.13) gives
pjk urj(t) wrj (t) + j
t0
Zrj (s)ds wrj (t). (5.3)
Also, if Qr(t) = 0, then by (2.14) jJ
urj(t) = wrq(t).
Hence by (5.3) for t [0, T] urj(t) br()T
(5.4)
if Qr(t) = 0.Now assume that Qr(t) > 0. Also assume there
exists 0 < u < t such that Qr(u) = 0 (we
comment on the case Qr(u) > 0 for all u [0, t] below) and
define
s = sup{u < t : Qr(u) = 0}. (5.5)
First assume that 0 < s < t (we comment on the case s = t
below). Note that Qr() > 0 for all
(s, t], because it is right continuous by definition, and so
Zr(u) = 0 for all u
[s, t] by (3.5).
By (2.13)
Zrj (t) Zrj (s) = arj(t) arj(s)
Drj (t) Drj (s)
jts
Zrj (s)ds
+pj
urj(t) urj(s)
. (5.6)
We show below that for any 0 < u < T
urj(u) urj(u) 2br()T
+Ar()
T
. (5.7)
This inequality also implies by (2.13) and the fact that Zj
D[0, T],Zrj (u) Zrj (u) 3br()
T+Ar()
T
. (5.8)
Note that because Zrj (s) = 0, (5.8) implies thatZrj (s)
3br()T
+Ar()
T
.
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Because Zrj (u) = 0 for all u (s, t]
j
ts
Zrj (s)ds = 0. (5.9)
Also by (5.4) and the fact that urj has left limits we
haveurj(s) br()T
. (5.10)
By combining (5.6)(5.10), for some CT > 0pjurj(t) Zrj (s) + 2
arj(t) + 2 Drj (t) +pj urj(s) CTbr()T
+Ar()
T
.
This inequality gives the desired result for ur (when there
exists 0 < u < t such that Qr(u) = 0).If Qr(u) > 0 for all
u > 0, the result follows by setting s = 0 in (5.6) and from
(2.6). Ifs = t in(5.5), the result follows from (5.7).
The result for Zr
follows from the bound for ur, (2.13) and Gronwalls inequality
(see Corollary
11.2 in [23]). The result for Qr follows trivially from the
bounds for ur and (2.14).To prove (5.7), we note that the number of
customers whose service started in pool j (including
those whose service times are equal to zero) in an interval [s,
t] is bounded by the summation of thenumber of customers whose
service is completed by that server pool and the total number of
arrivalsto the system in that interval. This follows from (2.7),
(2.8) and the fact under an admissible policyAj is non-decreasing.
Therefore
Arj(t) Arj(s) Ar(t) Ar(s) + Drj (t) Drj (s) +Arj(t)
n=Arj(s)+1
(1 j(n)),
which implies
Arj(t)
n=Arj(s)+1
j(n) Ar(t) Ar(s) + Drj (t) Drj (s).
This inequality gives (5.7) combined with (2.11) and the fact
that
r1/2
A
rj (t)
n=Arj(s)+1
j(n)
= r1/2 Arj(t) Arj(s) + arj(t) arj(s),
which follows from (2.10).
6 Proof of Theorem 4
The proof follows similarly to Theorem 3.2 in [29] using Lemma 7
(see also Step 1 of Proposition 1in [5]).
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Proof. Assume that (2.1), (2.2), (2.5) and (2.6) hold. We mainly
focus on the case where Assump-tion 1(i) holds. Fix a sequence of
admissible policies r. All the queueing processes below dependon
the policy r but we drop it from our notation for simplicity. We
show below that for any T > 0
Qr(t),Zr(t)
r T 0, as r
. (6.1)
Let br be defined as in (5.1). By (6.1), Theorem 4.4 in [10] and
random time change theorem (seeTheorem 5.3 in [11])
br B as r , (6.2)
where B is defined as in Section 4.Let (Zr, Qr, ur) be defined
as in (5.2). (We note that the process br depends on the policy
,
hence so is (Zr, Qr, ur).) Also set
Yr(t) = Qr(t) +J
j=1
Zr
j(t). (6.3)
By Theorem 3(i), under Assumption 1(i) for any T > 0
Yr(t) Yr(t) for all t [0, T]. (6.4)
Result (4.1) follows from (6.2), (6.4) and continuous mapping
theorem. Result (4.2) follows from(4.1) and Fatous lemma. Part (ii)
of Theorem 4 follows similarly using Theorem 3(ii) instead
ofTheorem 3(i) to arrive at
Yr(t)
+
Yr(t)
+
for all t [0, T]
instead of (6.4) and proceeding in a similar way as above.We
finish the proof by establishing (6.1) using Lemma 7. Fix T > 0.
Recall the definition
of wj and wrq in (2.12). By functional strong law of large
numbers (see Theorem 5.10 in [11]),Ar(t)
T 0 a.s. as r . Therefore by Lemma 7, it is enough to show
that
(wr, wrq) 0, as r , (6.5)
where wr = wr/
r and wrq = wrq/
r. For wrq , the result immediately follows from (2.6) and
the
fact that Ar is a delayed renewal process, see (2.4). Next we
focus on wr.By (2.12),
wrj (t) =Zr
j(0)
r + arj(t)/r Drj (t)/r. (6.6)
We have Zrj (0)/
r 0 as r by (2.6). By (2.10)
arj(t)/
r =Crj
rArj(t)
r
rpjArj(t).
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Because Arj(t) Ar(t) for all t and Ar(t) 2t a.s. for r large
enough, for any T > 0
sup0tT
Crj
rArj(t)
r
rpjArj(t) sup
0tT
Crj (2rt)
rpj2rt
(6.7)
for r large enough. The term on the RHS of (6.7) converges to 0
a.s. for any T > 0 as r byfunctional strong law of large numbers
(see Theorem 5.10 in [11]). Because Zrj (t) Nj/r jr,see (2.2), we
also have that sup0tT
Drj (t)/r 0 for any T > 0 by functional strong law oflarge
numbers, completing the proof of (6.1) by (6.6).
7 Proof of Theorem 5
We first establish two results and then prove Theorem 5 using
these results.
Proposition 9. Assume that (2.1), (2.2), (2.5) and (2.6) hold.
Under , for any 0 < s < T and > 0,
limsupr
P
Jj=2
Zrj ()[s,T]
>
= 0.
Proof. Assume that (2.1), (2.2), (2.5) and (2.6) hold. Fix T
> 0. As in [27] we argue below thatfor any > 0 and 0 < s
< T
lim supr
P
Jj=2
Zrj ()[s,T]
> 2
limsupr
P
supst1t2T|t2t1|
. (7.1)
Let Zr(t) = Zrj (t)/
r. By (2.13),
Zrj (t) = Zrj (0) + r
1/2arj(t) r1/2Drj (t) jt0
Zrj (s)ds + r1/2pju
rj(t).
By (6.1) and (6.5), this equation implies that
r1/2pjurj 0 as r .
Hence, for any T > 0, by (2.11)
sup0tT
Arj(t) jmj t 0 as r .17
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Therefore by the random time change theorem (see Theorem 5.3 in
[11])
arj aj, as r , (7.2)where aj is a driftless Brownian motion with
variance (1 pj)jj . Also by (6.1)
D
r
j
Dj, as r , (7.3)where Dj is a driftless Brownian motion with
variance jj. In addition
Ar A, as r , (7.4)where A is a Brownian motion with variance 2.
We get the desired result from (7.1), (7.2)(7.4),the fact that >
0 can be taken to be arbitrarily small and because Brownian motion
is continuousa.s.
We prove (7.1) next. First assume that
J
j=2
Zrj (0) = 0.
Let
r1 = inf{t > 0 :J
j=2
Zrj (t) > 2}
and
r0 = sup{r1 > t > 0 :J
j=2
Zrj (t) < }.
Recall thatZr
j (t) 0 for all t by (2.15). Also by (2.17), on [r
0 , r
1 ],J
j=2
pjurj(
r1 )
Jj=2
pjurj(
r0) Ar(r1 ) Ar(r0) +
Nr1rm1
(r1 r0 ) + (r1 r0 ).
Therefore, by (2.13)
Jj=2
Zrj (r1 )
Jj=2
Zrj (r0)
Jj=2
arj(r1 )
Jj=2
arj(r0)
J
j=2
Drj (r1 )
J
j=2
Drj (r0
) + Ar(r1 )
Ar(r0
) +
Nr1rm
1
(r1
r0 ) + (
r1
r0 ).(7.5)
This inequality gives (7.1).Now assume that
Jj=2
Zrj (0) < 0.
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Let
r = inf
t 0 :
Jj=2
Zrj (t) = 0
.
Note that the argument (7.5) still holds if we focus on [r
, T] instead of [0, T], therefore it is enoughto show that r 0
as r . This follows by noticing that (7.5) also holds if we set r1
= r anduse 0 instead of r0.
Proposition 10 (Convergence). Assume that (2.1), (2.2), (2.5)
and (2.6) hold. Under, for any0 < s < T
sups
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Similarly, by (3.3)(3.5)
Yr1 (t)
+= Qr(t) and
Yr1 (t)
= Zr1(t)/p1. (7.11)
By (2.13) and (2.14)
Yr1 (t) = wrq(t) +
wr1(t)
p1 1
p1
t
0Zr1(s)ds
Jj=2
urj(t). (7.12)
By (3.1) and (3.2)
Yr1 (t) = wrq(t) +
wr1(t)
p1 1
p1
t0
Zr1(s)ds J
j=2
urj(t). (7.13)
By (7.9)(7.12), and Gronwalls inequality (see (11.2) in
[23])
Yr1 (t) Yr1 (t) J
j=2
urj(t) urj(t) + C1 t
0
urj(s) urj(s) ds (7.14)
for C1 =1p1
exp
1p1
T
. First note that urj(0) = 0 andurj(0) Jj=1 Zrj (0). Hence,
(7.14)
implies (7.7), by (7.9)(7.11), Lemma 7 and (6.2).
We are now ready to prove Theorem 5.
Proof. Assume that (2.1), (2.2), (2.5) and (2.6) hold. We prove
the result using (7.6) and Propo-sition 11 below. We only prove the
result under Assumption 1(i), proof under Assumption 1(ii)
issimilar.
Fix > 0 and x > 0. By Proposition 11 we can find > 0
such that for r large enough
P
0
Yr(s)ds >
< and P
0
Y(s)ds >
< . (7.15)
Also, by (7.6) and the continuity of the integral operator we
have
P
T
Yr(s)ds > x
PT
Y(s)ds > x
,
as r . By (7.15), for x > 0 and r large enough
PT0
Yr
(s)ds > x P
T
Yr
(s)ds > x
+
PT
Y(s)ds > x
+ 2 P
T0
Y(s)ds > x
+ 3.
Since > 0 is arbitrary this proves (4.7) by Theorem 4. The
result (4.8) follows from (4.7) and theuniform integrability of
0 Y
r(s)ds by Proposition 11 below.
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Proposition 11. Assume that (2.1), (2.2), (2.5) and (2.6) hold.
For any admissible policy andT > 0
EQr()2T
E
Zr()2T
Eur()2T CT and
EQr()2T EZr()2T Eur()2T CTfor some CT > 0, independent of the
policy.
Proof. The first result follows from Lemma 7 and combining (52)
in [15] with Lemmas 2 and 3 in[5] and the fact that interarrival
times are assumed to have finite mean and variance. The
secondresult follows from Lemma 8 and a similar argument.
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