Manyserver queues with customer abandonment: numerical analysis of their diffusion models
ABSTRACT We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
firstinfirstout (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for manyserver queues, sometimes
for queues with as few as twenty servers.

Article: Validity of heavytraffic steadystate approximations in manyserver queues with abandonment
[Show abstract] [Hide abstract]
ABSTRACT: We consider GI/Ph/n+M parallelserver systems with a renewal arrival process, a phasetype service time distribution, n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the HalfinWhitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy traffic and steady state limits for GI/Ph/n+M queues.Queueing Systems 06/2013; · 0.44 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the FCFS $GI/GI/n$ queue in the HalfinWhitt heavy traffic regime, and prove bounds for the steadystate probability of delay (s.s.p.d.) for generally distributed processing times. We prove that there exists $\epsilon > 0$, depending on the interarrival and processing time distributions, such that the s.s.p.d. is bounded from above by $\exp\big(\epsilon B^2\big)$ as the associated excess parameter $B \rightarrow \infty$; and by $1  \epsilon B$ as $B \rightarrow 0$. We also prove that the tail of the steadystate number of idle servers has a Gaussian decay, and use known results to show that our bounds are tight (in an appropriate sense). \\\indent Our main proof technique is the derivation of new stochastic comparison bounds for the FCFS $GI/GI/n$ queue, which are of a structural nature, hold for all $n$ and times $t$, and build on the recent work of \citet{GG.10c}.06/2013;  SourceAvailable from: Shuangchi He[Show abstract] [Hide abstract]
ABSTRACT: We use an OrnsteinUhlenbeck (OU) process to approximate the queue length process in a $GI/GI/n+M$ queue. This onedimensional diffusion model is able to produce accurate performance estimates in two overloaded regimes: In the first regime, the number of servers is large and the mean patience time is comparable to or longer than the mean service time; in the second regime, the number of servers can be arbitrary but the mean patience time is much longer than the mean service time. Using the diffusion model, we obtain Gaussian approximations for the steadystate queue length and the steadystate virtual waiting time. Numerical experiments demonstrate that the approximate distributions are satisfactory for queues in these two regimes. To mathematically justify the diffusion model, we formulate the two overloaded regimes into an asymptotic framework by considering a sequence of queues. The mean patience time goes to infinity in both asymptotic regimes, whereas the number of servers approaches infinity in the first regime but does not change in the second. The OU process is proved to be the diffusion limit for the queue length processes in both regimes. A crucial tool for proving the diffusion limit is a functional central limit theorem for the superposition of timescaled renewal processes. We prove that the superposition of $n$ independent, identically distributed stationary renewal processes, after being centered and scaled in both space and time, converges in distribution to a Brownian motion as $n$ goes to infinity.12/2013;
Page 1
Stochastic Systems
arXiv: 1104.0347
MANYSERVER QUEUES WITH CUSTOMER
ABANDONMENT: NUMERICAL ANALYSIS OF THEIR
DIFFUSION MODELS∗
By Shuangchi He and J. G. Dai
Georgia Institute of Technology
We use multidimensional diffusion processes to approximate the
dynamics of a queue served by many parallel servers. The queue is
served in the firstinfirstout (FIFO) order and the customers wait
ing in queue may abandon the system without service. Two diffusion
models are proposed in this paper. They differ in how the patience
time distribution is built into them. The first diffusion model uses
the patience time density at zero and the second one uses the en
tire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distri
bution of such a diffusion process. A crucial part of the algorithm is
to choose an appropriate reference density. Using a conjecture on the
tail behavior of a limit queue length process, we propose a system
atic approach to constructing a reference density. With the proposed
reference density, the algorithm is shown to converge quickly in nu
merical experiments. These experiments also show that the diffusion
models are good approximations for manyserver queues, sometimes
for queues with as few as twenty servers.
1. Introduction.
multidimensional diffusion processes that approximate the dynamics of a
queue with many parallel servers. A manyserver queue serves as a building
block modeling operations of a largescale service system. Such a service
system may be a call center with hundreds of agents, a hospital depart
ment with tens or hundreds of inpatient beds, or a computer cluster with
many processors. When the customers of a service system are human be
ings, some of them may abandon the system before their service begins.
The phenomenon of customer abandonment is ubiquitous because no one
would wait for service indefinitely. As argued in Garnett, Mandelbaum and
The focus of this paper is the numerical analysis of
∗Supported in part by NSF Grants CMMI0727400, CMMI0825840, and CMMI
1030589.
AMS 2000 subject classifications: Primary 60K25, 60J70, 65R20; secondary 68M20,
90B22
Keywords and phrases: diffusion process, stationary distribution, phasetype distribu
tion, manyserver queue, heavy traffic, customer abandonment, quality and efficiency
driven regime
1
arXiv:1104.0347v2 [math.PR] 8 Apr 2011
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S. HE AND J. G. DAI
Reiman (2002), one must model customer abandonment explicitly in order
for an operational model to be relevant for decision making. We model cus
tomer abandonment by assigning each customer a patience time. When a
customer’s waiting time for service exceeds his patience time, he abandons
the queue without service.
The exact analysis of such a manyserver queue has been largely limited
to an M/M/n+M model (also called an ErlangA model) that has a Poisson
arrival process and exponential service and patience time distributions. See,
e.g., Garnett, Mandelbaum and Reiman (2002). However, as pointed out by
Brown et al. (2005), the service time distribution in a call center appears to
follow a lognormal distribution. Such distributions have also been observed
by Shi et al. (2010) for lengths of stay in a hospital. Moreover, the patience
time distribution in a call center has been observed to be far from exponential
by Zeltyn and Mandelbaum (2005). With a general service or patience time
distribution, there is no finitedimensional Markovian representation of the
queue. Except computer simulations, there is no method to exactly analyze
such a queue either analytically or numerically. To deal with the challenge,
the following strategies are adopted in this paper for analyzing a manyserver
queue.
First, the service time distribution is restricted to be phasetype. Since
phasetype distributions can be used to approximate any positivevalued
distribution, such a queueing model is still relevant to practical systems.
We focus on a GI/Ph/n + GI queue with n identical servers. The first GI
indicates that the customer interarrival times are independent and identi
cally distributed (iid) following a general distribution, the Ph indicates that
the service times are iid following a phasetype distribution, and the +GI
indicates that the patience times are iid following a general distribution. Sec
ond, we are particularly interested in a queue operating in the Quality and
EfficiencyDriven (QED) regime: The queue has a large number of servers
and the arrival rate is high; the arrival rate and the service capacity are
approximately balanced so that the mean waiting time is relatively short
compared with the mean service time. As argued in Garnett, Mandelbaum
and Reiman (2002), such a system has high server utilization as well as
short customer waiting times and a small fraction of abandonment. There
fore, both quality and efficiency can be achieved in this regime. Third, rather
than analyzing the manyserver queue itself, we propose and analyze diffu
sion models that approximate the queue. Two diffusion models are proposed
in this paper. In each model, a multidimensional diffusion process is used to
represent the scaled customer numbers among service phases. The difference
between the two diffusion models lies in how the patience time distribution
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NUMERICAL ANALYSIS OF DIFFUSION MODELS
3
is built into them. The first diffusion model uses the patience time density at
zero and the second one uses the entire patience time distribution. In partic
ular, the diffusion process in the first model is a multidimensional piecewise
OrnsteinUhlenbeck (OU) process. We propose an algorithm in this paper
to numerically solve the stationary distribution of a diffusion process. The
computed stationary distribution is used to estimate the performance mea
sures of a manyserver queue. Numerical examples in Section 6 demonstrate
that the diffusion models are very accurate in predicting the performance of
a manyserver queue, even if the queue has as few as twenty servers.
Except for the onedimensional case, the stationary distribution of a piece
wise OU process has no explicit formula. The algorithm proposed in this
paper is a variant of the one in Dai and Harrison (1992), which computes
the stationary distribution of a semimartingale reflecting Brownian motion
(SRBM). As in Dai and Harrison (1992), the starting point of our algorithm
is the basic adjoint relationship that characterizes the stationary distribution
of a diffusion process. With an appropriate reference density, the algorithm
can produce a stationary density that satisfies this relationship.
We set up a Hilbert space using the reference density. In this space, the
stationary density is orthogonal to an infinitedimensional subspace H. A
finitedimensional subspace Hkis used to approximate H and a function or
thogonal to Hkcan be numerically computed by solving a system of finitely
many linear equations. This function is used to approximate the stationary
density. There are two sources of error in computing the approximate sta
tionary density by our algorithm: approximation error and roundoff error.
The approximation error arises because Hkis an approximation of H. As
Hkincreases to H, the approximation error decreases to zero. The roundoff
error occurs because the solution to the system of linear equations has error
due to the finite precision of a computer. As Hkincreases to H, the dimension
of the linear system gets higher and the coefficient matrix becomes closer
to singular. As a consequence, the roundoff error increases. The condition
number of the matrix is used as a proxy for the roundoff error. Balancing
the approximation error and the roundoff error is an important issue in our
algorithm.
A properly chosen reference density is essential for the convergence of the
algorithm. By convergence, we mean that the approximation error converges
to zero as Hkincreases to H. More importantly, a “good” reference density
can make Hkconverge to H quickly so that the resulting approximation error
and roundoff error are small simultaneously even though the dimension of
Hkis moderate. To ensure the convergence of the algorithm, the reference
density should have a comparable or slower decay rate than the stationary
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S. HE AND J. G. DAI
density. Since the stationary density is unknown, we make a conjecture on
the tail behavior of the limit queue length process of manyserver queues with
customer abandonment. We conjecture that the limit queue length process
has a Gaussian tail and the tail depends on the service time distribution
only through its first two moments. This tail is used to construct a product
form reference density. With this reference density, the algorithm appears
to converge quickly, producing stable and accurate results. For comparison
purposes, we also test the algorithm with a certain “naively” chosen reference
density in Section 7.1. The algorithm fails to converge with the “naive”
reference density. The major contributions of this paper are the proposed
diffusion models and the proposed reference densities that are critical to the
numerical algorithm for computing the stationary distribution of a diffusion
model.
Our diffusion models are obtained by replacing certain scaled renewal
processes by Brownian motions. The replacement procedure is rooted in the
manyserver heavy traffic limit theorems that are proved in an asymptotic
regime. The two diffusion model proposed in this paper are motivated by the
diffusion limits proved in Dai, He and Tezcan (2010) and Reed and Tezcan
(2009). See Section 4.3 for more details. The theory of diffusion approxi
mation for manyserver queues can be traced back to the seminal paper by
Halfin and Whitt (1981), where a diffusion limit was established for GI/M/n
queues. Garnett, Mandelbaum and Reiman (2002) proved a diffusion limit
for M/M/n+M queues that allows for customer abandonment, and Whitt
(2005) generalized the result to G/M/n+M queues. Puhalskii and Reiman
(2000) established a diffusion limit for GI/Ph/n queues. Their result was
extended to G/Ph/n + GI queues with customer abandonment in Dai, He
and Tezcan (2010). Recently, Reed and Tezcan (2009) proved a diffusion
limit for GI/M/n + GI queues. In their framework, a refined limit process
is obtained by scaling the patience time hazard rate function.
Harrison and Nguyen (1990) derived Brownian models for multiclass open
queueing networks. Their diffusion models are SRBMs and are rooted in the
conventional heavy traffic limit theorems that are pioneered in Iglehart and
Whitt (1970) for serial networks and Reiman (1984) for singleclass networks.
See Williams (1996) for a survey of limit theorems in literature. For a two
dimensional SRBM living in a rectangle, Dai and Harrison (1991) proposed
an algorithm computing its stationary distribution. Dai and Harrison (1992)
extended the algorithm for an SRBM living in an orthant. To deal with the
unbounded state space, the notion of a reference density was first introduced
there. Their finitedimensional space Hkis constructed via (global) multi
nominals of order at most k. With this choice of Hk, the algorithm appears
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NUMERICAL ANALYSIS OF DIFFUSION MODELS
5
numerically unstable occasionally. In such a case, the roundoff error may
dominate the approximation error while the approximation error is still sig
nificant. Shen et al. (2002) extended Dai and Harrison (1991) to a hypercube
state space of an arbitrary dimension. They used a finite element method
to construct Hkto avoid numerical instability. Their algorithm sometimes
converges slowly because they did not explore a reference density. A linear
programming algorithm for computing the stationary distribution of a diffu
sion process was proposed in Saure, Glynn and Zeevi (2009). Both SRBMs
in an orthant and a diffusion approximation of manyserver queues with
two priority classes were investigated in their paper. Like the role of the
reference density, it appears that the rescaling of variables is essential to the
convergence of their algorithm.
The remainder of the paper is organized as follows. General diffusion pro
cesses are introduced in Section 2, where the basic adjoint relationship for a
diffusion process is also presented. In Section 3, we begin with recapitulat
ing the generic algorithm of Dai and Harrison (1992), and then propose a
finite element implementation that follows Shen et al. (2002). Two diffusion
models for GI/Ph/n + GI queues are presented in Section 4. In Section 5,
we discuss how to choose an appropriate reference density exploiting the
tail behavior of a diffusion process. In Section 6, it is demonstrated via nu
merical examples that the diffusion models serve as good approximations of
manyserver queues. Section 7 is dedicated to some implementation issues
arising from the proposed algorithm. The paper is concluded in Section 8.
We leave the proofs of Propositions 2 and 3 to the appendix.
Notation.
positive integers, real numbers, and nonnegative real numbers, respectively.
For d,m ∈ N, Rddenotes the ddimensional Euclidean space and Rd×m
denotes the space of d×m real matrices. We use C2
realvalued functions on Rdthat are twice continuously differentiable with
bounded first and second derivatives. For z,w ∈ R, we set z+= max{z,0},
z−= max{−z,0}, and z ∧ w = min{z,w}. All vectors are envisioned as
column vectors. For a ddimensional vector x ∈ Rd, we use xj for its jth
entry and diag(x) for the d×d diagonal matrix with jth diagonal entry xj.
For a matrix M, M?denotes its transpose, Mij denotes its (i,j)th entry,
and M = (?
its jth entry one and all other entries zero. Given two functions ϕ and ˆ ϕ
from N to R, we write ˆ ϕ(n) = O(ϕ(n)) as n → ∞ if there exists a constant
κ > 0 and some n0∈ N such that ˆ ϕ(n) ≤ κϕ(n) for all n > n0.
The symbols N, R, and R+ are used to denote the sets of
b(Rd) to denote the set of
i,jM2
ij)1/2. We reserve I for the d × d identity matrix, e for
the ddimensional vector of ones, and ejfor the ddimensional vector with
Page 6
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S. HE AND J. G. DAI
2. Diffusion processes.
on a ddimensional diffusion process X = {X(t) : t ≥ 0}. Let (Ω,F,F,P) be
a filtered probability space with filtration F = {Ft: t ≥ 0}. We assume that
X satisfies the following stochastic differential equation
?t
where the drift coefficient b is a function from Rdto Rd, the diffusion co
efficient σ is a function from Rdto Rd×m, and B = {B(t) : t ≥ 0} is an
mdimensional standard Brownian motion with respect to F. We assume
that both b and σ are Lipschitz continuous, i.e., there exists a constant
c1> 0 such that
Let d be a positive integer. This paper focuses
(2.1)X(t) = X(0) +
0
b(X(s))ds +
?t
0
σ(X(s))dB(s),
(2.2)
b(x) − b(y) + σ(x) − σ(y) ≤ c1x − y
for all x,y ∈ Rd.
Under condition (2.2), the stochastic differential equation (2.1) has a unique
strong solution, i.e., there exists a unique process X on (Ω,F,F,P) such that
(a) X is adapted to F, (b) for each sample path ω ∈ Ω, X(t,ω) is continuous
in t, and (c) for each t ≥ 0, the stochastic differential equation (2.1) holds
with probability one. See Øksendal (2003) for more details. We also assume
that σ is uniformly elliptic, i.e., there exists a constant c2> 0 such that
(2.3)y?Σ(x)y ≥ c2y?yfor all x,y ∈ Rd,
where
(2.4)Σ(x) = σ(x)σ?(x).
We are interested in the diffusion processes that model the dynamics of a
queue with many parallel servers. Parallelserver queues will be introduced in
Section 4. In that section, two diffusion processes will be identified to model
such a queue and the coefficients b and σ will be mapped out explicitly
in terms of primitive data of the queue. The diffusion models presented in
Section 4 are rooted in manyserver heavy traffic limit theorems proved in
Dai, He and Tezcan (2010) and Reed and Tezcan (2009).
A probability distribution π on Rdis said to be a stationary distribu
tion of X if X(t) follows distribution π for each t > 0 whenever X(0) has
distribution π. Condition (2.3) is required to ensure the uniqueness of the
stationary distribution. See Dieker and Gao (2011) for more details. In this
paper, we assume that X has a unique stationary distribution π and π has a
density g with respect to the Lebesgue measure on Rd. For a general diffusion
Page 7
NUMERICAL ANALYSIS OF DIFFUSION MODELS
7
process, there is no explicit solution for π. This paper develops a numerical
algorithm computing π. As in Dai and Harrison (1992), the starting point
of the algorithm is the basic adjoint relationship
?
where G is the generator of X defined by
(2.6)
d
?
(2.5)
RdGf(x)π(dx) = 0for all f ∈ C2
b(Rd),
Gf(x) =
j=1
bj(x)∂f(x)
∂xj
+1
2
d
?
j=1
d
?
?=1
Σj?(x)∂2f(x)
∂xj∂x?
for each f ∈ C2
b(Rd)
and Σ is the covariance matrix given by (2.4). The following theorem is a
consequence of Proposition 9.2 in Ethier and Kurtz (1986).
Theorem 1.
Then, π is a stationary distribution of X.
Let π be a probability distribution on Rdthat satisfies (2.5).
In this paper, we conjecture that a stronger version of Theorem 1 is true.
Conjecture 2.
and π(Rd) = 1. Then, π is a nonnegative measure and consequently it is a
stationary distribution of X.
Let π be a signed measure on Rdthat satisfies (2.5)
Our algorithm is to construct a function g on Rdsuch that
(2.7)
?
Assuming that Conjecture 2 is true, g must be the unique stationary density
of X. As a special case, the nonnegativity of a signed measure π that satisfies
(2.5) for a piecewise OU process was proposed as an open problem by Dai and
Dieker (2010). Piecewise OU processes will be introduced in Section 4.3.1.
Rdg(x)dx = 1and
?
RdGf(x)g(x)dx = 0for all f ∈ C2
b(Rd).
3. A finite element algorithm for stationary distributions.
this section, we propose a numerical algorithm computing the stationary
density g. The basic algorithm follows the one developed in Dai and Harrison
(1992). The finite element implementation closely follows Shen et al. (2002).
In
3.1. A reference density.
a notion called a reference density that was first introduced by Dai and
To compute the stationary density g, we adopt
Page 8
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S. HE AND J. G. DAI
Harrison (1992). A reference density for g is a function r defined from Rdto
R+such that
?
where
q(x) =g(x)
r(x)
(3.1)
Rdr(x)dx < ∞
and
?
Rdq2(x)r(x)dx < ∞,
for each x ∈ Rd
is called the ratio function. Such a function r exists because r = g is a
reference density. The reference density controls the convergence of our al
gorithm. We will discuss how to choose a reference density for the diffusion
models of a manyserver queue in Section 5.
For the rest of Section 3, we assume that a reference density r satisfying
(3.1) has been determined and remains fixed. In addition, we assume that
?
for j,? = 1,...,d. Since both b and σ are Lipschitz continuous, condition
(3.2) is satisfied if
?
Let L2(Rd,r) be the space of all squareintegrable functions on Rdwith
respect to the measure that has density r, i.e.,
?
where B(Rd) is the set of Borelmeasurable functions on Rd. Condition (3.1)
implies that q ∈ L2(Rd,r). We define an inner product on L2(Rd,r) by
?
The induced norm is given by
(3.2)
Rdb2
j(x)r(x)dx < ∞
and
?
RdΣ2
j?(x)r(x)dx < ∞
(3.3)
Rdx4r(x)dx < ∞.
L2(Rd,r) =f ∈ B(Rd) :
?
Rdf2(x)r(x)dx < ∞
?
?f,ˆf? =
Rdf(x)ˆf(x)r(x)dxfor f,ˆf ∈ L2(Rd,r).
(3.4)
?f? = ?f,f?1/2
for each f ∈ L2(Rd,r).
One can check that L2(Rd,r) is a Hilbert space and assumption (3.2) ensures
that Gf ∈ L2(Rd,r) for all f ∈ C2
relationship in (2.7) is equivalent to
b(Rd). In L2(Rd,r), the basic adjoint
(3.5)
?Gf,q? = 0for all f ∈ C2
b(Rd).
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NUMERICAL ANALYSIS OF DIFFUSION MODELS
9
With a fixed reference density r, we need only compute the ratio function
q by (3.5). Once q is obtained, we can compute the stationary density via
g(x) = q(x)r(x) for x ∈ Rd.
Let
(3.6)H = the closure of {Gf : f ∈ C2
b(Rd)}
where the closure is taken in the norm in (3.4). As a subspace of L2(Rd,r),
H is orthogonal to q. Let c be a constant function and c(x) = 1 for all
x ∈ Rd. Clearly, c ∈ L2(Rd,r) but c / ∈ H because
?
Let
(3.7)
?c,q? =
Rdg(x)dx = 1.
(3.8)¯ c = argmin
f∈H
?c − f?
be the projection of c onto H. Then, c−¯ c must be orthogonal to H. Assuming
that Conjecture 2 holds and X has a unique stationary density g, one must
have q = κq(c − ¯ c) for some constant κq ∈ R. By (3.7), the normalizing
constant κqsatisfies
κ−1
q
= ?c,c − ¯ c? = ?c − ¯ c,c − ¯ c? + ?¯ c,c − ¯ c? = ?c − ¯ c?2.
Hence, the ratio function is given by
(3.9)q =
c − ¯ c
?c − ¯ c?2.
3.2. An approximate stationary density.
first compute ¯ c, the projection of c onto H. The space H is linear and
infinitedimensional (i.e., a basis of H contains infinitely many functions).
In general, solving (3.8) in an infinitedimensional space is impossible. In
the algorithm, we use a finitedimensional subspace Hkto approximate H.
Suppose that there exists a sequence of finitedimensional subspaces {Hk:
k ∈ N} of H such that Hk→ H in L2(Rd,r) as k → ∞. Here, Hk→ H in
L2(Rd,r) means that for each f ∈ H, there exists a sequence of functions
{ϕk: k ∈ N} with ϕk∈ Hksuch that ?ϕk− f? → 0 as k → ∞. Let
¯ ck= argmin
f∈Hk
be the projection of c onto Hk. By Proposition 7 of Dai and Harrison (1992),
we have the following approximation result.
To compute q by (3.9), we need
?c − f?
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S. HE AND J. G. DAI
Proposition 1. Assume that Conjecture 2 is true. Then,
?qk− q? → 0as k → ∞,
where qk= (c−¯ ck)/?c − ¯ ck?2. Furthermore, when the reference density r is
bounded,
?
where gk(x) = qk(x)r(x) for each x ∈ Rd.
Rd(gk(x) − g(x))2dx → 0as k → ∞,
As in Dai and Harrison (1992), we choose
(3.10)Hk= {Gf : f ∈ Ck}
for some finitedimensional space Ck. In Section 3.3, we will discuss how to
construct Ckusing a finite element method. For notational convenience, we
omit the subscript k when k is fixed. The finitedimensional functional space
is thus denoted by C. Let mCbe the dimension of C and {fi: i = 1,...,mC}
be a basis of C. We assume that the family {Gfi: i = 1,...,mC} is linearly
independent in L2(Rd,r). Then,
(3.11)¯ ck=
mC
?
i=1
uiGfi
for some ui∈ R and i = 1,...,mC.
Using the fact ?Gfi,c − ¯ ck? = 0 for i = 1,...,mC, we obtain a system of
linear equations
(3.12)Au = v
where
(3.13)Ai?= ?Gfi,Gf??,u = (u1,...,umC)?,vi= ?Gfi,c?.
By the linear independence assumption, the mC× mCmatrix A is positive
definite. Thus, u = A−1v is the unique solution to (3.12). Once the vector u is
obtained, we can compute the projection ¯ ckby (3.11). Finally, the stationary
density g can be approximated via
g(x) ≈ gk(x) = r(x)c(x) − ¯ ck(x)
?c − ¯ ck?2
for each x ∈ Rd.
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NUMERICAL ANALYSIS OF DIFFUSION MODELS
11
3.3. A finite element method.
employed multinominals of orders up to k to construct the space Ck. This
choice appears to be numerically unstable. The approximation error is signif
icant when k is small, say, k ≤ 5. As k increases, the roundoff error in solv
ing (3.12) increases and ultimately dominates the approximation error. Al
though their implementation produces accurate estimates for the stationary
means of SRBMs, it sometimes produces poor estimates for the stationary
distributions. In this section, we construct a sequence of spaces {Ck: k ∈ N}
using the finite element method as in Shen et al. (2002). Because the state
space in Shen et al. (2002) is bounded, neither a reference density nor state
space truncation is used there.
The state space of X is unbounded in our setting. It is necessary to
truncate the state space to apply the finite element method. Let {Kk: k ∈
N} be a sequence of compact sets in Rd. For each f ∈ Ck, we assume that
f(x) = 0 for x ∈ Rd\ Kk. The subscript k is omitted again when it is
fixed and we use K to denote the compact support of the space C. In our
implementation, we restrict K to be a ddimensional hypercube
In Dai and Harrison (1992), the authors
(3.14)K = [−ζ1,ξ1] × ··· × [−ζd,ξd],
where both ζjand ξjare positive constants for j = 1,...,d.
We partition K into a finite number of subdomains. Such a partition is
called a mesh and each subdomain is called a finite element. Since K is
a hypercube, it is natural to use a lattice mesh, where each finite element
is again a hypercube. In this case, each corner point of a finite element is
called a node. In dimension j = 1,...,d, we divide the interval [−ζj,ξj] into
njsubintervals by partition points
−ζj= y0
j< y1
j< ··· < ynj
j
= ξj.
Then, K is divided into?d
ordinate (yi1
j=1nj finite elements. For future reference, we
label the nodes in the way that node (i1,...,id) corresponds to spatial co
1,...,yid
d), and define
h?
j= y?+1
j
− y?
j
for ? = 0,...,nj− 1 and j = 1,...,d.
If ∆ denotes such a mesh, we define
∆ = max{h?
j: ? = 0,...,nj− 1; j = 1,...,d}
and
(3.15) η∆= max
?h?1
j1
h?2
j2
: ?1,?2= 0,...,nj− 1; j1,j2= 1,...,d; j1?= j2
?
.
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12
S. HE AND J. G. DAI
The finitedimensional space C is generated using the above mesh. We
use the cubic Hermite basis functions to construct a basis of C, as in Shen
et al. (2002). The onedimensional Hermite basis functions for −1 ≤ z ≤ 1
are given by
(3.16)φ(z) = (z − 1)2(2z + 1)andψ(z) = z(z − 1)2.
In dimension j = 1,...,d and for ? = 1,...,nj− 1, let
φ?
j(z) =
φ
?z − y?
?z − y?
j
h?−1
j
?
?
if y?−1
j
≤ z ≤ y?
j,
φ
j
h?
j
if y?
j≤ z ≤ y?+1
j
,
0otherwise
and
ψ?
j(z) =
h?−1
j
ψ
?z − y?
?z − y?
j
h?−1
j
?
if y?−1
j
≤ z ≤ y?
j,
h?
jψ
j
h?
j
?
if y?
j≤ z ≤ y?+1
j
,
0otherwise.
Let x = (x1,...,xd)?be a vector in K. At node (i1,...,id), the basis func
tions of C are the tensorproduct Hermite basis functions
(3.17)fi1,...,id,χ1,...,χd(x) =
d?
j=1
gij,χj(xj)
where χjis either 0 or 1 and
gij,χj(z) =
?
φij
ψij
j(z)
j(z)
if χj= 0,
if χj= 1.
Therefore, each node has 2dtensorproduct basis functions and the space C
has a total of
(3.18)mC= 2d
d?
j=1
(nj− 1)
basis functions.
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NUMERICAL ANALYSIS OF DIFFUSION MODELS
13
The space C is not a subspace of C2
basis functions in (3.16), the second order derivative of φ(z) is not defined
at z = −1 and 1, and the second order derivative of ψ(z) is not defined
at z = −1, 0, and 1. As a consequence, there exists f ∈ C for which Gf
is not defined on the boundaries of certain finite elements. Because such
boundaries have Lebesgue measure zero in Rd, for each f ∈ C, we can find
a sequence of functions {ϕi: i ∈ N} in C2
as i → ∞. Hence, Hk⊂ H still holds for each k.
For the linear system (3.12) to have a unique solution, the family of func
tions
b(Rd). For the onedimensional Hermite
b(Rd) such that ?Gϕi− Gf? → 0
{Gfi1,...,id,χ1,...,χd: ij= 1,...,nj− 1;χj= 0,1;j = 1,...d}
must be linearly independent in L2(Rd,r). The following proposition pro
vides sufficient conditions for the linear independence. Its proof can be found
in the appendix.
Proposition 2.
tions (2.2) and (2.3) holds and all entries of Σ are continuously differen
tiable. Assume that r(x) > 0 for all x ∈ Rd. Then, the family of functions
Let G be the generator of X in (2.6) such that condi
{Gfi1,...,id,χ1,...,χd: ij= 1,...,nj− 1;χj= 0,1;j = 1,...d}
is linearly independent in L2(Rd,r), where fi1,...,id,χ1,...,χdis the basis func
tion of C given by (3.17). Consequently, the solution to the linear system
(3.12) is unique.
Now let us consider a sequence of functional spaces {Ck: k ∈ N}. Let
∆kbe the mesh for constructing Ck. We assume that the mesh ∆k+1is a
refinement of ∆k, i.e., a node or an interelement boundary in ∆kis also a
node or an interelement boundary in ∆k+1. We further assume that such
refinements are regular, i.e., for each η∆kdefined in (3.15), the set {η∆k:
k ∈ N} is bounded. The next proposition, along with Proposition 1, justifies
the proposed algorithm for computing the stationary distribution. We leave
the proof of Proposition 3 to the appendix, too.
Proposition 3.
that each ∆k+1is a refinement of ∆kand the refinements are regular. Let
Kkbe the ddimensional finite hypercube that is the domain of ∆k, and Ckbe
the functional space generated by ∆kusing the tensorproduct Hermite basis
functions in (3.17). Let H be the infinitedimensional space in (3.6) and
Let {∆k: k ∈ N} be a sequence of lattice meshes such
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S. HE AND J. G. DAI
Hkbe the finitedimensional space in (3.10), where the generator G satisfies
(2.2) and (3.2). Assume that
∆k → 0and Kk↑ Rd
as k → ∞.
Then,
Hk→ Has k → ∞.
4. Diffusion models for manyserver queues.
introduce GI/Ph/n + GI queues and present two diffusion models for such
a queue with many servers. The two models differ in how the patience time
distribution is built into them. The patience time density at zero is used in
the first model, whereas the entire patience time distribution is used in the
second model.
In this section, we
4.1. GI/Ph/n + GI queues in the QED regime.
with many servers working in the QED regime. The QED regime will be
discussed shortly. In this queue, the service time distribution is restricted
to be phasetype. All positivevalued distributions can be approximated by
phasetype distributions.
Let p be a ddimensional nonnegative vector whose entries sum to one, ν
be a ddimensional positive vector, and P be a d × d substochastic matrix.
We assume that the diagonal entries of P are zero and P is transient, namely,
I−P is invertible. Consider a continuoustime Markov chain with d+1 phases
(or states) where phases 1,...,d are transient and phase d+1 is absorbing.
For j = 1,...,d, the Markov chain starts in phase j with probability pj. The
amount of time it stays in phase j is exponentially distributed with mean
1/νj. When it leaves phase j, the Markov chain enters phase ? = 1,...,d
with probability Pj?or enters phase d+1 with probability 1−?d
from starting until absorption in phase d+1 for the above Markov chain. In
particular, when P is a zero matrix, the associated phasetype distribution
is a hyperexponential distribution with d phases.
In a GI/Ph/n+GI queue, there are n identical servers working in parallel.
The customer arrival process is a renewal process. Upon arrival, a customer
enters service immediately if an idle server is available. Otherwise, he waits
in a buffer with infinite waiting room that holds a firstinfirstout (FIFO)
queue. The service times form a sequence of iid random variables, following
a phasetype distribution. When a server finishes serving a customer, the
server takes the leading customer from the waiting buffer. When the buffer
is empty, the server begins to idle. Each customer has a patience time. The
We focus on a queue
?=1Pj?. The
phasetype distribution with parameters (p,ν,P) is the distribution of time
Page 15
NUMERICAL ANALYSIS OF DIFFUSION MODELS
15
patience times are iid following a general distribution. When a customer’s
waiting time in queue exceeds his patience time, the customer abandons the
system with no service.
Let λ be the arrival rate and 1/µ be the mean service time. The system
is assumed to operate in the QED regime, i.e., both the arrival rate λ and
the number of servers n are large, while the traffic intensity ρ = λ/(nµ) is
close to one. Because customer abandonment is allowed, it is not necessary
to assume ρ < 1 for the system to reach a steady state. For future purposes,
we put
β =√n(1 − ρ).
(4.1)
Assume that the phasetype service time distribution has parameters
(p,ν,P). Each service time can be decomposed into a number of phases.
When a customer is in service, he must be in one of the d phases. Let Zj(t)
denote the number of customers in phase j service at time t. In steadystate,
one expects that the customers in service are distributed among the d phases
following a distribution γ, given by
(4.2)
One can check that?d
Suppose that all customers, including those initial customers waiting in
the buffer at time zero, sample their first service phases following distribution
p upon arrival. One can stratify customers in the waiting buffer according
to their first service phases. For j = 1,...,d, we use Wj(t) to denote the
number of waiting customers at time t whose service begins with phase j.
Then,
γ = µR−1p andR = (I − P?)diag(ν).
j=1γj= 1 and γjis interpreted to be the fraction of
phase j service load on the n servers.
(4.3)Yj(t) = Zj(t) + Wj(t)
is the number of phase j customers in the system, either waiting or in service.
Let Y (t) be the corresponding ddimensional random vector and
(4.4)
˜Y (t) =
1
√n(Y (t) − nγ).
In each diffusion model, the process˜Y = {˜Y (t) : t ≥ 0} is approximated by
a ddimensional diffusion process.
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