MATHEMATICS OF OPERATIONS RESEARCH
Vol. 23, No. 2, May 1998
Printed in U.S.A.
POLLING SYSTEMS IN HEAVY TRAFFIC:
A BESSEL PROCESS LIMIT
E. G. COFFMAN, JR., A. A. PUHALSKII,
AND
M. I. REIMAN
This paper studies the classical polling model under the exhaustive-service assumption; such
models continue to be very useful in performance studies of computer/communication systems.
The analysis here extends earlier work of the authors to the general case of nonzero switchover
times. It shows that, under the standard heavy-traffic scaling, the total unfinished work in the
system tends to a Bessel-type diffusion in the heavy-traffic limit. It verifies in addition that, with
this change in the limiting unfinished-work process, the averaging principle established earlier by
the authors carries over to the general model.
1. Introduction. In classical polling models, M ¢ 2 queues are visited by a single
server in cyclic order. Such models have many applications in the performance analysis
of communication systems, including token rings and packet switches, where a singleserver resource (e.g., a communication link) is shared among many demands on the
resource (e.g., traffic streams). An analysis of the 5ESSt switching system performed by
Leung (1991) is a modern example and a sequel to work of Kruskal (1969) on earlier
switching systems. Introductions to a massive literature addressing many different applications can be found in Takagi (1986, 1990) and Levy and Sidi (1990).
This paper focuses on polling with exhaustive service: The visit of the server to any
given queue terminates only when no work remains to be done at that queue. We number
the queues from 1 to M and assume they are served in that order. The time for the server
to switch over (or move) from queue i to queue i / 1 is nonzero in general, and is allowed
to be random and to depend on i.
An exact analysis of exhaustive polling systems is quite difficult; hopes for explicit
solutions are soon abandoned in favor of numerical methods and approximations. A recent
study of asymptotic behavior derived from heavy-traffic (diffusion) limits has been a
promising approach, one that leads to relatively simple formulas which in turn yield useful
insights. The cornerstone of the theory is an averaging principle proved in Coffman,
Puhalskii, and Reiman (1995). In a recent application of this principle, Reiman and Wein
(1998) study set-up scheduling problems in two-class single-server queues. Olsen (1995)
provides a heuristic refinement of the averaging principle that improves the quality of the
resulting approximation for waiting time distributions in moderate loading.
A limitation of the results in Coffman et al. (1995) is the often untenable assumption
of zero switchover times. The main contribution of this paper is a proof that the total
unfinished work in the general two-queue system tends, in the heavy-traffic limit, to a
Bessel type diffusion rather than the reflected Brownian motion in the case of zero switchover times. We verify that, as a corollary, the averaging principle in Coffman et al. (1995)
carries over to the general model. The remainder of this section describes the averaging
principle and gives a heuristic argument leading to the new diffusion limit for exhaustive
polling systems. Section 2 introduces notation and formulates our main results. A threshold
queue very similar to the one in Coffman et al. (1995) is analyzed in §3. Results for this
Received October 24, 1995; revised: December 18, 1996; October 9, 1997.
AMS 1991 subject classification. Primary: 60K25, 90B22.
OR/MS Index 1978 subject classification. Primary: Queues/Polling systems.
Key words. Polling systems, heavy traffic limit, averaging principles, Bessel process, semi-martingales.
257
0364-765X/98/2302/0257/$05.00
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queueing system supply bounds for the polling system which lead to the averaging principle, as shown in §4. Further preliminaries are taken up in §5, where the tightness of a
number of basic processes is proved. The development of §§3–5 culminates in the proofs
in §6 of our main results. A critical element in the proofs is a semimartingale representation
of the unfinished work process which allows us to use general convergence results
for semimartingales from Jacod and Shiryaev ( 1987 ) and Liptser and Shiryaev (1989).
Briefly, the mathematical model is as follows. Customers arrive at the ith queue in a
renewal process with rate li and interarrival-time variance s 2ai . The service rate parameter
at the ith queue is mi and the service-time variance is s 2si . Let di be the mean switchover
time from queue i to queue i / 1. Define r Å r1 / ··· / rM , where ri Å li / mi is the
traffic intensity at queue i.
We first review the case of M Å 2 queues and zero switchover times di Å 0, 1 ° i
° M, adapting the presentation of Coffman et al. (1995), which was for queue lengths,
to the context of unfinished work. Let Ut , t ¢ 0, denote the total unfinished work (service
time) in queues 1 and 2 at time t. Then since the process (Ut , t ¢ 0) is the same as the
unfinished work process in the corresponding SGI /G/1 system, we can extend the heavytraffic limit theorem of Iglehart and Whitt (1970) as follows (see also Reiman (1988)).
Consider a sequence of systems indexed by n, and let r n denote the traffic intensity of
the nth system. The heavy-traffic limit stipulates that r n r 1 as n r ` with
q_
n ( r n 0 1) å c n r c as n r ` , 0` õ c õ ` .
(As in the standard set-up, we also assume that lni r li ú 0, ( s nsi ) 2 r s si2 as n r ` , i
Å 1, 2. There is one more technical assumption that we defer until later; it implies that
the Lindeberg condition holds.) For the scaled process V nt Å n 01 / 2U ntn , n ¢ 1, 0 ° t ° 1,
d
under the above conditions, V n r V, as n r ` , where V is reflected Brownian motion
with infinitesimal drift c and variance
M
s2 å
∑ li ( s si2 / r 2i s 2ai ) ú 0.
iÅ1
The averaging principle proved in Coffman et al. (1995) deals with queue lengths; converted to unfinished work, the principle states that, for any continuous function f : R/
r R and any T ú 0, we have
*
T
(1.1)
d
f (V tn,i )dt r
0
* S * f (uV )duD dt,
T
1
t
0
i Å 1, 2,
0
where
V n,i
t is the time scaled and normalized unfinished work at queue i and the symbol
d
r denotes convergence in distribution. Extended to general M, the corresponding averaging principle for sojourn times Wt (i) at queue i is given by
* f (Y
T
(1.2)
n,i
t
d
)dt r
0
* * f S 1 0◊ r V uD dudt,
T
1
i
t
0
1 ° i ° M,
0
q_
n
where Y n,i
t Å W nt (i)/ n , n ¢ 1, 0 ° t ° 1, and ◊ Å (1°j õk°M rj rk .
We now return to nonzero switchover times with the expected values di , 1 ° i ° M.
While a similar averaging principle can be expected, the unfinished-work process is no
longer the same as in the SGI/G/1 system, so the limit diffusion V may be different. To
see what this limit process should be, we give the following heuristic argument. The
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259
purpose of the remainder of the paper is to formulate the argument precisely and to prove
rigorously that it is correct.
Consider the same sequence of systems as before, assuming in addition to the previous
conditions that d ni r di , 0 ° di õ ` . As before, V nt Å n 01 / 2U nnt . The drift c(x ) of the
limit process V at point x is the limit D r 0, n r ` of
n
c nD (x) Å D 01 E[V t/D
0 V tn ÉV tn Å x ú 0].
Work enters the system at rate r n per unit time. We assume that D is small enough that
V nt does not reach zero during [t, t / D]. With nonzero switchover times, work leaves
the system at a rate less than 1, which we calculate as follows. Let r n (x) denote the
V nt Å x . Then
fraction of time the server spends doing useful work (not switching) when
q_
n
r ( x) is the rate at which work leaves the system. Since there are O( n ) cycles per unit
of ‘‘diffusion’’ time, we can write
r n (x) Å
E[useful work done over a cycle]
.
E[duration of a cycle]
For simplicity, let M Å 2 andqstart
_ the cycle at the moment the server switches to queue
1.
q_ On average, it takes time nx/(1 0 r1 ) to empty queue 1, d1 to switch to queue 2,
nx/(1 0 r2 ) to empty queue 2, and d2 to switch back to queue 1. The useful work is
q_
q_
nx
nx
v(x) Å
/
,
1 0 r1 1 0 r 2
so we have
r n (x) Å
v(x)
.
v(x) / d1 / d2
But in heavy traffic, r1 / r2 Å 1, so a little algebra yields
r n (x) Å
(1.3)
x
q_ ,
x / d/ n
where d Å r1r2 (d1 / d2 ). Extending the calculation to general
q_ M gives the same result
with d generalized to d Å ◊(d1 / ··· / dM ). Now c nD (x) Å n [ r n 0 r n (x )], so the limit
D r 0, n r ` yields
c nD (x) r c(x) å c / d/x.
Note thatqthe
_ seemingly innocuous addition of a O(1) switchover time to a cycle which
takes O( n ) time (before normalization) produces a dramatic change in the form of
the drift.
A heuristic calculation along the above lines shows that the infinitesimal variance is
unaffected by the addition of switchover times. We are thus led to expect that V n r V,
where the limit process V is a one-dimensional diffusion with state dependent drift c(x ),
and constant variance s 2 . This fact is proved rigorously for M Å 2. The limit process is
a Bessel process with negative drift. When 2d/ s 2 õ 1, the process can hit the origin, in
which case it instantaneously reflects. When c õ 0, V is positive recurrent and has a
stationary distribution with density
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
p(x) Å
(1.4)
a(ax) be 0ax
,
G( b / 1)
x ¢ 0,
where a Å 2É cÉ/ s 2 , b Å 2d/ s 2 . This is the gamma density of order b and scale a.
We further verify that the averaging principles (1.1) and (1.2) hold for M Å 2, with V
the above Bessel process. Extended to general M, we have
1
T
*
T
d
f (Y n,i
t )dt r
0
1
T
* * f S 1 0◊ r V uD dudt,
T
1
i
t
0
0
so if we let V0 have the stationary distribution (1.4), then
E
1
T
**
T
0
1
0
f
S
D
1 0 ri
Vtu dudt Å
◊
*
`
0
a(ax) be 0x
G( b / 1)
F* S
1
f
0
D G
1 0 ri
ux du dx.
◊
For example, if f ( x ) Å x, we find that the limiting sojourn times have the means
b / 1 1 0 ri
.
a
2◊
2. Results. We begin with notational matters. In the standard set-up for heavy traffic
limits, we consider a sequence of two-queue polling systems. For the nth system, denote
n,l
n,l
by t n,l
i Å j 1 / ··· / j i , i ¢ 1, l Å 1, 2, the time of the ith arrival to the lth queue in
terms of interarrival times j n,l
i ,
h n,l
i , i ¢ 1, l Å 1, 2, the ith service time in the lth queue,
s n,l
i , i ¢ 1, l Å 1, 2, the ith switchover time from the lth queue.
n,l
n,l
We assume that j n,l
i , i ¢ 1, h i , i ¢ 1, s i , i ¢ 1, l Å 1, 2, are independent i.i.d.
n,l
sequences, and that j i ú 0, i ¢ 1. As in the previous section, we introduce, for n Å 1,
2, . . . and l Å 1, 2,
01
lnl Å (Ej n,l
,
1 )
r nl Å
mln Å (Eh 1n,l ) 01 ,
lln
,
mln
d ln Å Es n,l
1 ,
r n Å r 1n / r 2n .
Instead of dealing with the variances ( s nai ) 2 and ( s sin ) 2 , it is more convenient here to
introduce
n n,l 2
( s nl ) 2 Å E( h n,l
1 0 rl j1 ) ,
l Å 1, 2.
As in §1, we assume the limits, as n r ` for l Å 1, 2,
(2.1)
lnl r ll ,
mnl r ml ú 0,
s nl r sl ú 0,
d ln r dl ,
(2.2)
and assume the heavy traffic condition
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q_
lim n ( r n 0 1) Å c.
(2.3)
nr`
The Lindeberg conditions mentioned earlier are, for l Å 1, 2, e ú 0,
q_
2
n,l
(2.4)
lim E( j n,l
1 ) ·1( j 1 ú e n ) Å 0,
nr`
q_
2
n,l
lim E( h n,l
1 ) ·1( h 1 ú e n ) Å 0,
(2.5)
nr`
q_
2
n,l
lim E(s n,l
1 ) ·1(s 1 ú e n ) Å 0.
(2.6)
nr`
Also let
s 2 Å l1s 21 / l2s 22 ú 0,
(2.7)
rl Å ll / ml , l Å 1, 2.
Recall that U nt is the total unfinished work in the nth system at time t, with U n0 independent
n,l
n,l
of { j n,l
i , i ¢ 1}, { h i , i ¢ 1}, and {s i , i ¢ 1}, l Å 1, 2, and that
1
V nt Å q_ U nnt ,
n
(2.8)
t ¢ 0,
V n Å (V nt , t ¢ 0).
Recalling that d Å r1r2 (d1 / d2 ), let the process X Å ( Xt , t ¢ 0) solve the equation
(2.9)
dXt Å [2(d / c(Xt Û 0) 1 / 2 ) / s 2 ]dt / 2s(Xt Û 0) 1 / 2 dWt ,
X0 ¢ 0,
where W Å (Wt , t ¢ 0) is a standard Brownian motion, and X0 and W are independent.
Next, define V Å (Vt , t ¢ 0) as the diffusion process on [0, ` ) with the generator
S D
Lg(x) Å c /
d
x
dg
1
d2g
(x) / s 2 2 (x),
dx
2
dx
where the domain of L is
D(L) Å {g √ C 2K ([0, ` )) : g(x) Å g̃(x 2 )
for some g̃ √ C 2K ([0, ` ))},
C 2K ([0, ` )) being the space of twice continuously differentiable functions on [0, ` ) with
compact support.
A proof of the following technical result is similar to the proof of the existence of the
Bessel diffusion (Ikeda and Watanabe (1989), Chapter 4, Examples 8.2 and 8.3).
LEMMA 2.1. For given
__exist and are unique in law.
q_ _ X0 and V0 , the processes X and qV
If V0 is distributed as X0 , then the distributions of V and X coincide .
In the main result below, and throughout the remainder of the paper, all processes are
assumed to have right-continuous with left-hand limits sample paths and considered as
random elements of the Skorohod space D[0, ` ) (see, e.g., Jacod and Shiryaev (1987),
Liptser and Shiryaev (1989)), and convergence in distribution for the processes is und
derstood as weak convergence of the induced measures on D[0, ` ). By r we denote
P
convergence in distribution in an appropriate metric space. Also, r denotes convergence
in probability.
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
d
THEOREM 2.1. Assume that V n0 r V0 , as n r ` , where V0 is a nonnegative random
variable. If conditions (2.1) – (2.6) hold, then
d
V n r V.
n,l
Theorem 2.1 allows us to get the averaging principle for unfinished work. LetqU
_t ,t
n,l
n,l
¢ 0, denote the unfinished work at time t at queue l Å 1, 2, and define V n,l
t Å U nt / n ,V
n,l
Å (V t , t ¢ 0).
THEOREM 2.2. Let f ( x), x ¢ 0, be a real-valued continuous function. Then, under
the conditions of Theorem 2.1, for t ú 0,
*
t
d
f (V sn,l )ds r
0
* S * f(uV )duD ds,
t
1
l Å 1, 2.
s
0
0
To conclude this section, it is instructive to compare the result of Theorem 2.1 with a
related Bessel process limit obtained by Yamada (1984, Theorem 1). (An example of
this type for point processes is considered by Yamada (1986). Rosenkrantz (1984) considers an alternative approach to the problem studied in Yamada (1984).) Note that the
process of total unfinished work satisfies the equation
* 1(U
t
U nt Å U n0 / S nt 0
(2.10)
n
s
ú 0) a ns ds,
0
where
A n,l
t
(2.11)
n
t
S ÅS
n,1
t
/S
n,2
t
,
S
n,l
t
Å
∑ h in,l , l Å 1, 2,
iÅ1
A n , l Å (A n,l
t , t ¢ 0), l Å 1, 2, are the input processes, i.e.,
S
D
j
A tn,l Å max j : ∑ j in,l ° t ,
iÅ1
and a ns is the indicator of the event that the server is not switching over (i.e., is serving)
at time s.
According to (2.10), if U ns ú 0, then the instantaneous rate at which work leaves the
system is a ns . The heuristic argument of §1 shows that it is reasonable to replace a ns by
˘ tn , t ¢ 0) defined as the solution to
r n (U sn ), i.e., consider the process Ŭ n Å (U
* 1(U˘
t
(2.12)
˘ tn Å U 0n / S tn 0
U
n
s
˘ sn )ds
ú 0)r n (U
0
as an approximation for U n . Equation (2.12) is of the type studied by Yamada. The
conditions of our Theorem 2.1 allow us, with some reservations, to apply his Theorem 1;
the limit process that this gives us turns out to be the same as the one in Theorem 2.1.
This comparison justifies our guess that r n (U ns ) can be substituted for a ns in (2.10).
Moreover, it is plausible to conjecture that one can weaken the much more restrictive
conditions of Yamada’s result. Indeed, the techniques developed in the proof of Theorem
2.1 can be applied to prove the following generalization of Yamada’s result. In this gen-
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263
˘ nt , t ¢ 0) is a nonnegative process satisfying (2.12),
eralization, we assume that Ŭ n Å (U
n
where r (x), x ¢ 0, is
q_a nonnegative bounded function, not necessarily from (1.3). We
˘ nnt / n , t ¢ 0, V̆ n Å (V˘ nt , t ¢ 0) and rV n Å supx¢0 r n (x ). The previous
further let V˘ nt Å U
notation is preserved.
THEOREM 2.3.
Assume that r n (x ) satisfies the following conditions:
(r1) lim x(rV n 0 r n (x)) Å d,
x,nr`
(r2) sup x(rV n 0 r n (x)) õ ` .
x,n
q_
Assume that, as n r ` , n ( r n 0 rV n ) r c and conditions (2.1), (2.4) and (2.5) hold. If
d
d
V̆ n0 r V0 , then V̆ n r V.
The main improvements over Yamada’s result are that we do not need the input processes to be Poisson (Yamada conjectured that this extension holds, but did not give a
proof) and that we do not assume the condition r n ° rV n . In addition, r n ( x) does not have
to be nondecreasing, the initial condition U n0 does not need a second moment and the
increments of S nt do not need fourth moments.
3. A Threshold Queue. In this section we prove an averaging principle for a singleserver queue, called the threshold queue, which is central to our analysis. The threshold
queue is basically the standard FIFO single-server queue described in Coffman et al.
(1995) except that the threshold operates on the unfinished work, not the queue length.
For a given parameter h ¢ 0, busy periods of the threshold queue begin only when the
unfinished work first exceeds h; busy periods terminate in the normal way, when no
unfinished work remains. We say that the server switches on when the busy periods begin
and switches off when the busy periods end. Those periods during which the server is
switched off are called accumulation periods; such a period includes the usual idle period
plus a period during which arrivals are accumulating in the queue. An accumulation period
and its following busy period make up a cycle.
Threshold queues correspond in the obvious way to the queues in our two-queue polling
system; for example, the accumulation periods of the threshold queue representing queue
1 correspond to the busy periods of queue 2. In our general approach to the proof of the
averaging principle (cf. Theorem 2.2), the time interval [0, T] is divided into subintervals
sufficiently small that the total unfinished work in the system remains approximately
constant during each. Then, during a subinterval, the behavior of the unfinished work at
each queue is approximated by that of a threshold queue. The main result of this section
(Theorem 3.1) shows that a threshold queue also obeys an averaging principle; the averaging principle for the polling system is derived as a consequence of the averaging
principles for the threshold queues defined for the subintervals.
We use the notation of Coffman et al. (1995). Consider a sequence of threshold queues
indexed by n. The generic interarrival and service times are denoted by j nq_and h n respectively. The threshold for the unfinished work in the nth queue is h n Å nan , where
a n is a given constant. We are assuming that
(3.1)
sup E( j n ) 2 õ ` ,
n
sup E( h n ) 2 õ ` ,
n
and, letting l n Å (Ej n ) 01 and mn Å (Eh n ) 01 , assume that
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
(3.2)
lim l n Å l ú 0,
lim mn Å m ú 0,
lim a n Å a ú 0,
nr`
nr`
nr`
l õ m.
As in Coffman et al. (1995), within each busy period, at most one of the interarrival
periods is allowed to be exceptional, i.e., to have a distribution other than that of j n .
Specifically, for each i ¢ 1, we introduce a nonnegative random variable j̃ ni and an integervalued random variable x ni which correspond to the ith cycle. If there are at least x ni
arrivals in the ith busy period, then the ( x ni )th arrival has an exceptional interarrival
period whose duration is taken to be j̃ ni . If the busy period has less than x ni arrivals, no
exceptional arrivals occur. We assume that there exists a family of sequences { z ni (r), i
¢ 1}, r ú 0, of identically distributed nonnegative random variables such that
(3.3)
1 n
P
q__ z 1 (r) r 0
n
q_
t n 
as n r ` , r ú 0, lim lim ∑ P(j˜ ni ú z ni (r)) Å 0,
rr` nr`
t ú 0,
iÅ1
and that the joint distribution of z ni (r), the normal interarrival times, and the service times
in the ith cycle does not depend on i. We allow for two interarrival times to be dependent
if one is taken from a busy period of the ith cycle and the other is taken either from
another cycle or from an accumulation period of the ith cycle. However, interarrival
(except for the exceptional), as well as service, times within each accumulation or busy
period are assumed to be mutually independent. We also assume that the time of the first
arrival, which we denote by jV n1 , may have a distribution different from that of the generic
interarrival time, and that
jV 1n
q_
(3.4)
P
r 0.
n
_
q
Introduce Xn (t) Å Y n (nt)/ n , t ¢ 0, where Y n (t) is the unfinished work at t, and assume
that X n (0) Å 0.
The following result is well known and will be used several times in the remainder of
the paper (see Iglehart and Whitt (1970) for a proof).
LEMMA 3.1. Let { z ni , i ¢ 1}, n ¢ 1, be a triangular array of nonnegative i.i.d.
random variables such that, for any e ú 0,
q_
lim E( z n1 ) 2·1( z n1 ú e n ) Å 0.
nr`
Let N n , n ¢ 1, be nonnegative integer-valued random variables such that, for some q
ú 0,
S
lim P
nr`
D
Nn
ú q Å 0.
n
Then as n r ` ,
1
q_
n
P
max z ni r 0.
1°i°N n
THEOREM 3.1. Let f ( x), x √ R/ , denote a bounded continuous function . If conditions
(3.1) – (3.4) hold, then for any T ú 0
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*
T
1
P
f ( X n (t))dt r T
0
PROOF .
the times
* f (au)du
265
as n r ` .
0
We proceed as in the proof of Theorem 3.1 in Coffman et al. (1995). Define
g n0 Å 0,
a ni Å inf(t ú g ni01 : X n (t) ú 0),
i ¢ 1,
(3.5)
b ni Å inf(t ú g ni01 : X n (t) ú a n ),
g ni Å inf(t ú b ni : X n (t) Å 0),
i ¢ 1,
i ¢ 1.
Note that the b ni start and the g ni terminate busy periods. We prove that
g n
(3.6)
P
q_
nt 
ra
and
*
q_
g n nt
(3.7)
P
S
D
1
1
/
mt
l m0l
S
f ( X n (s))ds r mt
0
1
1
/
l m0l
as n r ` ,
D*
a
f (u)du
as n r ` ,
0
which immediately give the assertion of the theorem.
n
For i ¢ 2, denote by jV ni the time between g ni01 and the first arrival after g i01
, i.e., jV ni
n
n
n,1
n,2
Å a i 0 g i01 ; and denote by { j i ,k , k ¢ 1} and { j i ,k , k ¢ 1} the i.i.d. sequences, with
generic random variable j n , from which normal interarrival times on [ a ni , b ni ] and [ b ni ,
a ni/1 ], respectively, are taken. Similarly, let { h n,l
i ,k , k ¢ 1}, i ¢ 1, l Å 1, 2, be the sequences
from which service times of requests arriving in [ a ni , b ni ] and [ b ni , g ni ], respectively, are
n,l
drawn. Note that, by the conditions of the theorem, the distribution of { z ni (r), j n,l
i ,k , h i ,k ,
l Å 1, 2, k ¢ 1} does not depend on i Å 1, 2, . . . .
In a sense, the jV ni , i ¢ 2, also represent exceptional interarrival times. By Lemma 3.1
in Coffman et al. (1995), we know that they satisfy conditions similar to those imposed
on j̃ ni , i.e.,
q_
t n 
(3.8)
lim lim ∑ P(jV ni ú zV ni (r)) Å 0,
rr` nr`
t ú 0,
iÅ2
where
zV in (r) Å
max
q_
1°k°  r n 
n,2
j i01,k
,
i ¢ 2,
r ú 0.
Moreover, as n r ` ,
zV ni (r) P
r 0,
q_
n
(3.9)
i ¢ 2,
r ú 0.
Define for i ¢ 1
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
`
S
D
k
n,1
Vn
Vn
A n,1
,
i (t) Å 1(j i ° t) / ∑ 1 j i / ∑ j i , j ° t
(3.10)
kÅ1
S
x ni01
jÅ1
D S∑
D
k
A n,2
∑ j in,2, j ° t / 1
i (t) Å ∑ 1
kÅ1
jÅ1
(3.11)
S
`
/
D
x ni01
˜ ni ° t
j n,2
i, j / j
jÅ1
k
∑ 1 ∑ j in,2, j / j˜ in ° t ,
kÅx ni
jÅ1
k
n,l
S n,l
i (k) Å ∑ h i , j ,
(3.12)
k Å 1, 2, . . . ,
l Å 1, 2.
jÅ1
For homogeneity of notation, we further set zV n1 (r) Å jV n1 . As in Coffman et al. (1995), by
(3.3) and (3.8), it is enough to prove
(3.6) and (3.7) on the events
_
q
t n 
n
G (r) Å
>
iÅ1
{j˜ in ° z in (r), jV in ° zV in (r)}.
Define the interval lengths
u ni Å b ni 0 g ni01 ,
£ ni Å g ni 0 b ni ,
so that by (3.5) and (3.10) – (3.12)
(3.13)
H
H
i ¢ 1,
J
1
n,1
n
,
u ni Å inf t ú 0: q_ S n,1
i (A i (nt)) ú a
n
£ ni Å inf
1
1
n,1
n
t ú 0: q_ [nt 0 S in,2 (A in,2 (nt))] ú q_ S n,1
i (A i (nu i ))
n
n
J
,
and
g ni 0 g ni01 Å u in / £ in .
(3.14)
In analogy with (3.10) and (3.11), define (since r is fixed, it is omitted in the new notation
below)
`
S
D
D
k
D
n,1
Vn
Vn
AV n,1
,
i (t) Å 1(z i (r) ° t) / ∑ 1 z i (r) / ∑ j i , j ° t
kÅ1
`
S
S∑
k
jÅ1
n,1
A n,1
,
i (t) Å 1 / ∑ 1 ∑ j i , j ° t
kÅ1
(3.15)
`
AV n,2
i (t) Å 1 / ∑ 1
kÅ1
`
S
jÅ1
k
j n,2
,
i, j ° t
jÅ1
k
D
n
n,2
A n,2
,
i (t) Å ∑ 1 z i (r) / ∑ j i , j ° t
kÅ1
jÅ1
and define as in (3.13) and (3.14)
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POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
H
H
H
H
J
J
1
n
V n,1
,
uV ni Å inf t ú 0: q_ S n,1
i (A i (nt)) ú a
n
(3.16)
1
n,1
n
,
u ni Å inf t ú 0: q_ S n,1
i ( A i (nt)) ú a
n
£V ni Å inf
£ ni Å inf
1
1 n,1 V n,1
n
V n,2
t ú 0: q_ [nt 0 S n,2
i (A i (nt))] ú q_ S i (A i (nuV i ))
n
n
J
J
,
1
1 n,1 n,1
n,2
n
t ú 0: q_ [nt 0 S n,2
i ( A i (nt))] ú q_ S i ( A i (nu i ))
n
n
,
and
(3.17)
gV ni Å
i
i
jÅ1
jÅ1
∑ (uV nj / £V nj ), g ni Å ∑ (u nj / £ nj ),
gV 0n Å g 0n Å 0.
i ¢ 1,
Note that since u ni , uV ni and u ni are defined in terms of the same process S n,1
i , we actually
n
n
n,1
n
V n,1
have A n,1
i (nu i ) Å A i (nuV i ) Å A i (nu i ) so that
n,1
n
n,1
n
n,1
n,1
n
V n,1
S n,1
i (A i (nu i )) Å S i (A i (nuV i )) Å S i ( A i (nu i )).
q_
Since jV ni ° zV ni (r), j˜ ni ° z ni (r), 1 ° i ° t n , on G n ( r), we have by (3.10), (3.11) and
(3.15) that
(3.18)
n,1
n,1
AV n,1
i (t) ° A i (t) ° A i (t),
(3.19)
n,2
V n,2
A n,2
i (t) ° A i (t) ° A i (t),
q_
on G n ( r), and hence by (3.13), (3.16) and (3.18), for 1 ° i ° t n ,
u ni ° u ni ° uV ni , £ ni ° £ ni ° £V ni ,
q_
on G n ( r), and then by (3.14) and (3.17), for 1 ° i ° t n ,
(3.20)
(3.21)
g ni 0 g ni01 ° g ni 0 g ni01 ° gV ni 0 gV ni01 ,
on G n ( r).
_
_
Now we prove (3.6) for gV n nt  and g n nt  ; this will imply (3.6) for g n
_
Consider only the upper bound process. The proof for g n nt  is similar.
First, note that by (3.15),
q
q
q_
nt 
on G n (r).
q
S
D
k
AV n,1
k ¢ 0 : zV ni (r) / ∑ j n,1
,
i (t) Å inf
i, j ú t
jÅ1
S
k/1
D
AV n,2
k ¢ 0 : ∑ j n,2
/ 1.
i (t) Å inf
i, j ú t
jÅ1
n,l
Since { j n,l
i ,k , k ¢ 1}, { h i ,k , k ¢ 1}, l Å 1, 2, are i.i.d., we have by (3.1) and (3.2) that
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
q_
q_
1  nt  n,l P t
,
q_ ∑ j i ,k r
l
n kÅ1
1  nt  n,l P t
,
q_ ∑ h i ,k r
m
n kÅ1
l Å 1, 2,
and hence, by (3.3), (3.9), (3.12), and Lemma 2.1 in Coffman et al. (1995),
P l
1 n,l q_ P
1 n,l n,l q_
(3.22)
t,
l Å 1, 2.
q_ AV i ( nt) r lt,
q_ S i (AV i ( nt)) r
m
n
n
By Lemma 2.1 in Coffman et al. (1995) and (3.16), (3.2)
q_
P am
(3.23)
nuV ni r
.
l
q_ P
n
V n,1
Hence, by (3.22), S n,1
i (A i (nuV i ))/ n r a which gives us by Lemma 2.1 in Coffman et
al. (1995), (3.16), and (3.22)
q_
P
am
(3.24)
n £V ni r
,
i ¢ 1.
m0l
Then, by (3.17) and (3.23),
q_
P
1
1
(3.25)
n(gV ni 0 gV ni01 ) r am
/
l m0l
S
D
i ¢ 1.
,
Since
q_
gV
1  nt  q_
Å q_ ∑ n(gV nk 0 gV nk01 ),
n kÅ1
n q_
 nt 
and since gV ni 0 gV ni01 , i ¢ 1, are identically distributed by construction, we would have,
in view of Lemma 2.4 in Coffman et al. (1995),
gV n
(3.26)
q_
nt 
P
S
r am
D
1
1
/
t
l m0l
provided
q_
q_
lim lim nP( n (gV n1 0 gV n0 ) ú k) Å 0.
(3.27)
kr` nr`
By (3.17), this would follow from
q_
q_
lim lim nP( nuV n1 ú k) Å 0,
kr` nr`
(3.28)
q_
q_
lim lim nP( n £V n1 ú k) Å 0.
kr` nr`
Consider the first limit. By (3.16)
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269
q_
q_
q_
n
V n,1
P( nuV n1 ú k) Å P(S n,1
1 (A 1 ( nk)) ° na )
(3.29)
q_
q_
q_
q_
1 n
n,1 1 n
° P(AV n,1
(
nk)
°
l
nk)
/
P(S
(
l
nk)
°
na n ).
1
1
2
2
By (3.12) we have, applying Chebyshev’s inequality and (3.1) and (3.2), that, for any e
ú 0,
SZ
q_
lim lim nP
(3.30)
kr` nr`
D
Z
q_
q_
k q_
S n,l
(k
n)
0
n
¢
k
n e Å 0,
1
mn
l Å 1, 2.
By an analogue of (3.30) for interarrival times, and by (3.15), in analogy with the proof
of (3.20) in Coffman et al. (1995),
q_
q_
q_
q_
n
lim lim nP(ÉAV n,l
1 ( nk) 0 l k n É ¢ k n e) Å 0,
(3.31)
l Å 1, 2.
kr` nr`
Relations (3.29) – (3.31) prove the first convergence in (3.28).
For the second convergence, we first prove that
S
S
DD
q_
1
ln
lim lim nP w n1 ú
1 0 n k Å 0,
3
m
kr` nr`
(3.32)
where
1
n
n
V n,1
w n1 Å q_ S n,1
1 (A 1 (nuV 1 )) 0 a .
n
(3.33)
Note that w n1 is nonnegative. By the inequality
1
sup
h n,1
w n1 ° q_
1, j ,
n
1 ( nuV 1 )
n 1° j°AV n,1
we have
q_
S
S
DD
q_
q_
1
ln
1 0 n k ° nP( nuV n1 ú k)
3
m
_
_
q
q
q_
1
l n q_
n
n
n,1
/ nP(AV n,1
(
nk)
ú
m
nk)
/
m
nkP
h
ú
1
0
k n .
1
1,1
3
mn
q_
q_
We have proved that limkr` lim nP( nuV n1 ú k) Å 0; by (3.31), since l õ m, we have
nr`q_
q_
q_
n
limkr` lim nP(AV n,1
1 ( nk) ú m nk) Å 0. Finally, by Chebyshev’s inequality,
nP w n1 ú
nr`
S
nkP h n,1
1,1 ú
S
S
S
D D
D D
1
l n q_
9
n,1 2
10 n k n °
E( h 1,1
) ,
3
m
(1 0 l n / mn ) 2k
and applying (3.1) and (3.2), we arrive at (3.32).
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
Going back to the proof of the second inequality in (3.28), write, by (3.16) and (3.33),
in analogy with (3.29),
q_
q_
q_
q_
q_
n
n
V n,2
P( n £V n1 ú k) Å P(sup( nt 0 S n,2
1 (A 1 ( nt))) ° nw 1 / na )
t°k
q_
q_
q_
n
n
V n,2
° P(S n,2
1 (A 1 ( nk)) ¢ n (k 0 w 1 ) 0 na )
q_
q_
1 n
1
ln
n
n
° P(AV n,2
(
nk)
ú
(
l
/
m
)
nk)
/
P
w
ú
1
0
k
1
1
2
3
mn
q_
q_
1 n
2 1 ln
/ P S n,2
( l / mn ) nk ¢ n
/
k 0 an
.
1
2
3 3 mn
S
D SS
S S
S
D
DD
DD
Putting together (3.30), (3.31), (3.32) and the inequality l õ m yields the second convergence in (3.28). Thus, (3.26) has been proved. The proof of (3.6) is done.
To prove (3.7) on G n (r), we apply Lemma 2.4 in Coffman et al. (1995), i.e., we prove
that
SH Z *
q_
q_
1  nt 
lim q_ ∑ P
nr` n iÅ1
g ni
f ( X n (s))ds
n
n
g i01
(3.34)
S
0m
1
1
/
l m0l
D*
Z J
a
D
f (u)du ú e > G n (r) Å 0,
0
e ú 0,
and
SH Z *
q_
 nt 
lim lim ∑ P
(3.35)
kr` nr`
iÅ1
q_
g ni
n
g ni01
Z J
D
f ( X n (s))ds ú k > G n (r) Å 0.
Note that (3.35) is easy. For, by the right inequality in (3.21) and the boundedness of f,
we have, letting \·\ denote the sup norm,
q_
 nt 
SH Z *
q_
lim ∑ P
nr` iÅ1
g ni
n
g ni01
Z J
D
q_
 nt 
q_
n
f ( X (s))ds ú k > G (r) ° lim ∑ P( n (gV in 0 gV i01
) \ f \ ú k)
n
n
nr` iÅ1
which tends to 0 as k r ` by (3.27) and by the fact that the (gV ni 0 gV ni01 ), i ¢ 1, are
identically distributed.
By (3.14), (3.34) would follow if
q_
SH Z *
1  nt 
lim q_ ∑ P
nr` n iÅ1
q_
u ni
n
0
f ( X n (s / g ni01 ))ds 0
m
l
* f (u)duZ ú eJ > G (r) D Å 0,
a
n
0
and
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POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
SH Z *
q_
q_
1  nt 
lim q_ ∑ P
nr` n iÅ1
£ ni
f ( X n (s / b ni ))ds
n
0
(3.36)
0
m
m0l
* f (u)duZ ú eJ > G (r) D Å 0.
a
n
0
These limits have similar proofs; we prove only (3.36), which is more difficult.
First, by the second set of inequalities in (3.20) and the fact that {£V ni , i ¢ 1} and
{ £ ni q,_i ¢ 1} each consist of identically distributed random variables, for d ú 0, 1 ° i
° t n,
SH Z
Z J
q_
am
ú d > G n (r)
m0l
n£ ni 0
P
SZ
°P
D
Z D SZ
q_
am
úd /P
m0l
n £V n1 0
q_
n £ n1 0
Z D
am
úd ,
m0l
and hence
SH Z
q_
1  nt 
lim q_ ∑ P
nr` n iÅ1
(3.37)
q_
n£ ni 0
SZ
q_
° t lim P
nr`
Z J
am
ú d > G n (r)
m0l
n £V n1 0
D
Z D
am
ú d / t lim P
m0l
nr`
SZ
q_
n £ n1 0
Z D
am
ú d Å 0,
m0l
where the last equality follows by (3.24) and its counterpart for £ n1 . Next,
SH Z *
q_
P
£ ni
f ( X n (s / b in ))ds 0
n
0
SH *
q_
n£ni Úam/(m0l)
0
0f
u
q_
a0
10
q_
\ f \· n £ ni 0
l
u
m
/ b in
n
S S D DZ
SH Z
/P
ZS S
f Xn
°P
m
m0l
du ú
Z J
am
e
ú
m0l
2
* f (u)duZ ú eJ > G (r) D
a
n
0
DD
J
e
> G n (r)
2
> G n (r)
D
D
.
q_
q_
Sum the second term on the right over i Å 1, . . . ,  t n  and divide by n . By (3.37),
the result tends to 0 in probability as n r ` , so the proof of (3.36) will be finished by
proving
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
SH *
q_
1  nt 
lim q_ ∑ P
nr` n iÅ1
q_
n £ ni Úam/( m0l )
0
ZS S
DD
S S D DZ
f Xn
(3.38)
0f
u
q_
/ b in
n
a0 10
l
u
m
> G n (r)
D
Å 0.
ú h > G n (r)
D
Å 0.
du ú
e
2
J
We prove first that, for h ú 0,
SH
q_
1  nt 
lim q_ ∑ P
n
n iÅ1
q_
u°
sup
n£ni Ú am / ( m0 l )
Z S
Xn
D
S S D DZ J
u
q_
(3.39)
0
/ b in
n
a0 10
l
u
m
By construction,
X n (u / b ni ) Å a n / w in /
n,2
S n,2
i (A i (nu)) 0 nu
,
q_
n
u √ [0, £ in ],
where wni is defined in analogy with (3.33), so,
SH
P
q_
u°
sup
n £ niÚam / ( m0 l )
SH
S
°P
Z S
Xn
sup
u°am / ( m0 l )
/ P w ni ú
h
3
Z
u
q_
/ b in
n
D S S D DZ J
0
a0
10
l
u
m
ú h > G n (r)
q_
n,2
S n,2
(A
(
nu)) l
h
i
i
0 u ú
> G n (r)
q_
m
3
n
Z J
D S
/ 1 Éa n 0 a É ú
h
3
D
D
D
.
n,2
V n,2
Since the distributions of ( A n,2
i (t), t ¢ 0), (A i (t), t ¢ 0) and (S i (t), t ¢ 0) do not
depend on i, we conclude from (3.32), (3.2) and (3.19) that the left-hand side of (3.39)
is not greater than
S
t lim P
nr`
sup
u°am / ( m0 l )
Z
Z D
q_
l
h
n,2
V
S n,2
(A
(
nu)) 0 u ú
1
1
m
3
n
1
q_
/ t lim P
nr`
S
sup
u°am / ( m0 l )
Z
Z D
q_
l
h
n,2
S n,2
u ú
1 ( A 1 ( nu)) 0
m
3
n
1
q_
q_
q_
n,2
which is zero by (3.22) and an analogous relation for S n,2
1 ( A 1 ( nu))/ n ; (3.39) is
proved.
Now on the event
/ 3904 0014 Mp
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H
q_
u°
sup
n £ niÚam / ( m0 l )
Z S
Xn
u
q_
/ b in
n
D S S D DZ J
0
a0
l
u
m
10
°h ,
q_
q_
we haveqthat
_ X n (u/ n / b in ) ° a / h, u √ [0, n £ in Ú am/( m 0 l )], and therefore, for
u √ [0, n £ ni Ú am/( m 0 l )],
ZS S
f Xn
u
q_
DD S S D D Z
/ b in
0f
n
a0
10
l
u
m
° vf (h, a / h ),
where vf (d, T ) is the modulus of continuity of f on [0, T] for partitions of diameter d.
This implies by the continuity of f that, for all h small enough and for all i,
H
q_
u°
sup
n £ niÚam / ( m0 l )
Z S
Xn
H*
u
q_
/ b in
n
D S S D DZ J
0
ZS S
q_
,
a0 10
n£ ni Úam /( m0l )
f Xn
0
u
q_
l
u
m
/ b in
n
°h
DD S S D D Z
0f
a0 10
l
u
m
du °
e
2
J
,
so for h small enough
SH *
SH
q_
n£ in Úam/( m0 {l)
0
°P
ZS S
f Xn
P
q_
u°
sup
n£ niÚam / ( m0 l )
Z S
Xn
u
q_
/ b in
n
u
q_
/ b in
n
DD S S D D Z J D
D S S D DZ J D
0f
0
a0
a0
10
10
l
u
m
l
u
m
du ú
e
2
> G n (r)
ú h > G n (r)
,
and so (3.38) follows from (3.39). Thus (3.36), (3.34) and (3.7) are proved. This comh
pletes the proof of the theorem.
4. An averaging principle for the unfinished work. In this section, having in view
T
the averaging principle, we derive a limit theorem for the integral *0 f (V tn,1 )dt, where
d
f ( x) is a real-valued continuous function on the positive half-line, assuming that V n r Ṽ
for some continuous process Ṽ. This is carried out by providing suitable upper and lower
bounds for the unfinished work at an individual queue in analogy with the contents of §4
in Coffman et al. (1995). The main result is the following.
d
THEOREM 4.1. Assume that, in addition to the conditions of Theorem 2.1, V n r Ṽ,
where Ṽ Å (Ṽt , t ¢ 0) is a nonnegative continuous process such that, for any T ú 0,
*0T 1(Ṽt Å 0)dt Å 0 P-a.s. Then, for any continuous function f ( x ) on R/ ,
* f (V
T
0
n,1
t
d
)dt r
* S * f (uV˜ )duD dt.
T
1
t
0
0
PROOF . We first assume that f ( x) is bounded and nonnegative. We note that it is
enough to prove that
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
*
T
(4.1)
d
f (V tn,1 )·1( d ° V nt ° K)dt r
0
* S * f (uV˜ )duD·1( d ° V˜ ° K)dt,
T
1
t
0
t
0
for any d and K, 0 õ d õ K, such that
* [1(V˜ Å d ) / 1(V˜ Å K)]dt Å 0
T
(4.2)
t
t
P-a.s.
0
The argument is given in the proof of Theorem 2.1 in Coffman et al. (1995). So, we
prove (4.1) assuming (4.2). The idea of the proof is the same as in Coffman et al. (1995).
Note that if considered in isolation an individual queue in our polling system passes
through alternating periods of accumulating and serving requests, thus its behavior resembles the behavior of the threshold queue above, the distinction being that here the threshold
is a random process: the queue starts being served when the unfinished work at this queue
becomes equal to the total amount of the unfinished work in the system. According to the
assumptions of the theorem, the (properly normalized and time-scaled) process of the
total unfinished work is a continuous process in the limit. Therefore, we can divide the
time axis into (random) intervals small enough for the total unfinished work during an
interval to be close to a constant. Then during such an interval an individual-queue unfinished work is well approximated by the unfinished work in a threshold queue with the
associated constant as a threshold. The proof of the theorem implements this program.
As in Coffman et al. (1995), choose e √ (0, d /2) such that N Å (K 0 d )/ e is an integer
and, given r( e ) õ e /2, let, for 0 ° i ° N,
ai ( e ) Å d / ie,
Br ( e ) ( e, i) Å (ai ( e ) 0 r( e ), ai ( e ) / r( e )),
Cr ( e ) ( e, i) Å (0, ai ( e ) 0 e / r( e )) < (ai ( e ) / e 0 r( e ), ` ),
z 0n ( e, i) Å 0,
t nk ( e, i) Å inf(t ú z nk01 ( e, i) : V nt √ Br ( e ) ( e, i)),
z nk ( e, i) Å inf(t ú t nk ( e, i) : V nt √ Cr ( e ) ( e, i)),
k ¢ 1,
k ¢ 1,
z0 ( e, i) Å 0,
tk ( e, i) Å inf(t ú zk01 ( e, i) : V˜ t √ Br ( e ) ( e, i)),
zk ( e, i) Å inf(t ú tk ( e, i) : V˜ t √ Cr ( e ) ( e, i)),
k ¢ 1,
k ¢ 1.
Thus, [ t nk ( e, i), z nk ( e, i)] are intervals during which V nt ‘‘does not vary too much.’’
For the sequel, we note that, since Ṽ is continuous, the argument of the proof of Lemma
4.1 in Coffman et al. (1995) applies to V n and Ṽ to give that
tk ( e, i) õ zk ( e, i)
(4.3)
P-a.s.
on { tk ( e, i) õ ` },
lim P( min zk ( e, i) ° T ) Å 0,
kr`
0°i°N
and that r( e ) can be chosen so that, as n r ` ,
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(V n , ( t kn ( e, i) Ú T, z kn ( e, i) Ú T )k¢1,0°i°N )
(4.4)
d
˜ ( tk ( e, i) Ú T, zk ( e, i) Ú T )k¢1,0°i°N ),
r (V,
where convergence in distribution is in D[0, ` ) 1 R ` (Billingsley (1968)).
n
Let n k n,1
j (i, k), j ¢ 0, denote the successive times after n t k ( e, i), when the unfinished
work at queue 1 becomes equal to 0. These are times when switchovers from queue 1 to
queue 2 start. We also let q n,1
i ,k denote the number of accumulation-service cycles for queue
1 in [ t nk ( e, i) Ú T, z kn ( e, i) Ú T], i.e.,
q n,1
i ,k Å
H
n,1
min( j : k j/1
(i, k) ú z kn ( e, i) Ú T ),
if k 0n,1 (i, k) ° z kn ( e, i) Ú T,
0,
n
if k n,1
0 (i, k) ú z k ( e, i) Ú T.
We now define two threshold queues approximating queue 1 on [ t nk ( e, i) Ú T, z nk ( e,
i) Ú T] whose unfinished work processes bound the unfinished work process of queue 1
from below and from above, respectively. We begin by introducing a threshold queue
associated with queue 1 on [ t nk ( e, i) Ú T, z nk ( e, i) Ú T].
n
Fixing i and k, we denote k nj Å k n,1
j (i, k ) and let n u j , j ¢ 1, denote the successive
n
times after n k 0 when the server starts serving queue 1; obviously, k nj01 õ u nj õ k nj , j
¢ 1. Let the arrivals to queue 1 on [n k n0 , ` ) be numbered successively starting from 1.
Let j˜ n1 denote the time period between n k n0 and the first arrival. Denote by j˜ nl , l ¢ 2, the
times between the (l 0 1)th and lth of these arrivals. Obviously, {j˜ nl , l ¢ 2} is a sequence
of i.i.d. random variables with the distribution of the generic interarrival time for queue
1. Introduce independent replicas { j n,m
j ,l , l ¢ 1}, j ¢ 1, m Å 1, 2, of the interarrival time
sequence at queue 1 and independent replicas { h n,m
j ,l , l ¢ 1}, j ¢ 1, m Å 1, 2, of the
service time sequence at queue 1.
Given h ú 0, let
n
w nj Å inf(t ú k j01
: V tn,1 ú h) Ú u nj ,
j ¢ 1,
c jn Å inf(t ú u jn : V tn,1 ° V wn,1nj ),
j ¢ 1.
n
n
n
n
Note that if V n,1
u nj ° h, then w j Å c j Å u j . Let x j , j ¢ 1, index the last arrival to queue
n
n
1 occurring no later than at n c j and let £V j , j ¢ 1, denote the time between n c nj and the
( x nj / 1)th arrival. By definition, £V nj ° jV nx nj/1 .
q_
Construct as follows a threshold queue with the threshold h n Å nh. In the first cycle
the interarrival times in the accumulation period are taken from the sequence {j˜ n1 , j˜ n2 , . . . ,
n,1
n
j˜ nx n1 , j˜ nx n1/1 , j 1,1
, j n,1
1,2 , ···}. The associated service times for arrivals 1 through x 1 are
those of corresponding arrivals to queue 1 and the subsequent service times are h n,1
1,1 ,
h n,1
1,2 , . . . .
Denoting the threshold queue normalized and time-scaled unfinished work at t by
VV n,1
t , define
b 1n Å inf(t ú 0 : VV tn,1 ú h).
Then n b n1 ends the first accumulation period. Obviously, b n1 Å w n1 0 k n0 if V wn,1n1 ú h which
n,1
˜ nx n1/1 , j 1,1
happens if V n,1
, j n,1
u n1 ú h, in this case j
1,2 , ··· are not actually used as interarrival
n,1
n,1
times and h 1,1 , h 1,2 , ··· are not used as service times. At n b 1n service switches on. If
n,1
n
V n,1
w n1 ° h, which happens if V u n1 ° h, then interarrival times after n b 1 are taken from
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
n
n,2
n
{ j n,2
1,l , l ¢ 1} and service times after n b 1 are taken from { h 1,l , l ¢ 1} until time n f 1 ,
where
f n1 Å inf(t ú b n1 : VV n,1
° V n,1
t
w n1 ).
n,2
n,2
We then take j n,2
1,xV n1 to be the last random variable in the sequence { j 1,1 , j 1,2 , ···} that is
n
n
n,1
actually realized as an interarrival time in [n b 1 , n f 1 ]. In the case that V w n1 ú h, we define
n
f n1 Å b n1 and set xV n1 Å j n,2
1,xV n1 Å 0. In both cases, the first arrival after n f 1 is made to occur
n
n
n
at time n f 1 / £V 1 and bring the same service time as the ( x j / 1)th arrival in queue 1 so
n,2
˜n
that its interarrival time £ n1 satisfies the inequality £ n1 ° £V n1 / j n,2
1,xV n1/1 ° j x n1/1 / j 1,xV n1/1 .
The subsequent interarrival times are j˜ nx n1/2 , . . . , j˜ xn n2 , and the service times are the same
as for arrivals x n1 / 2, . . . , x n2 in queue 1, where x n2 is the index of the last arrival in
queue 1 occurring no later than at n c n2 . Arrivals x n2 / 1, x n2 / 2, ··· have the interarrival
n,1
n,1
n,1
times j˜ nx n2/1 , j n,1
2,1 , j 2,2 , ··· and the service times h 2,1 , h 2,2 , ··· until (and unless, i.e., these
n,1
times are not used if V w n2 ú h) the threshold has been exceeded. After this has happened
at n b n2 , where
b 2n Å inf(t ú g 1n : VV tn,1 ú h),
and
g n1 Å inf(t ú b n1 : VV n,1
Å 0),
t
and until n f n2 , where
f 2n Å inf(t ú b 2n : VV tn,1 ° V wn,1n2 ),
n,2
n,2
n,2
the interarrival times are j n,2
2,1 , j 2,2 , ··· and the service times are h 2,1 , h 2,2 , ··· (as above
n,1
n
n
n
these are not used if V w n2 ú h and hence f 2 Å b 2 ). After n f 2 , the next arrival brings the
same service time as the arrival in the original queue terminating the interarrival time
n,1
j˜ nx n2/1 and occurs at time n f 2n / £V 2n (in both cases V n,1
w n2 ú h and V w n2 ° h) so that its
n
n
n,2
n
n,2
˜
interarrival time satisfies £ 2 ° £V 2 / j 2,xV n2/1 ° j x n2/1 / j 2,xV n2/1 , where j n,2
2,xV n2 denotes the last
n,2
n
random variable from { j n,2
2,1 , j 2,2 , ···} that is realized as an interarrival time in [n b 2 ,
n
n
n,2
n,1
n
n f 2 ] (again xV 2 Å j 2,xV n2 Å 0 if V w n2 ú h). The subsequent interarrival times are j˜ x n2/2 , ···
and the service times replicate those of queue 1 until the unfinished work hits 0 after
which the cycle resumes.
That this is indeed a threshold queue with generic interarrival and service times distributed as in the original queue follows by Lemma 4.2 in Coffman et al. (1995). The
exceptional arrivals, if any, are the ones occurring at n f nj / £V nj , j ¢ 1. Note also that if
˜n
V n,1
u nj ° h, then j x nj/1 is used for constructing interarrival sequences in both accumulation
and busy periods so that these sequences, generally, are dependent. This explains why we
emphasised this assumption in Theorem 3.1.
We now check that VV n ,1 satisfies the conditions of Theorem 3.1. We need focus only
on the part related to exceptional interarrival times and the time of the first arrival. Define
z jn (r) Å
max
q_
1°k°  nr 
j˜ x̃n nj/k /
max
q_
1°k°  nr 
j jn,2
,k ,
where x̃ nj indexes the first arrival in the original queue after k nj01 . Noting that {j˜ nx˜ nj/k , k
¢ 1} is distributed as {j˜ nk , k ¢ 1} and that £ nj ° j˜ xn nj/1 / j n,2
j ,xV nj/1 , one can prove in analogy
with Lemma 3.1 in Coffman et al. (1995) that the sequence { £ nj , z nj (r), j ¢ 1} satisfies
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the conditions of Theorem 3.1. Informally, this follows by
the fact that since the number
q_
of arrivals in an accumulation-service
q_ cycle is of order n , ‘‘with high probability’’ the
exceptional arrival is among the first nr arrivals when r is ‘‘big,’’ similarly, j n,2
j ,xV nj/1 belongs
n,2
n,2
_
q
to the set { j j ,1 , . . . , j j, nr } ‘‘with high probability’’ for r ‘‘big.’’ By a similar argument,
q_
P
j̃ n1 / n r 0. Thus, the conditions of Theorem 3.1 hold for the associated threshold queue.
We now define a process Ṽ n ,1 as the process VV n ,1 corresponding to the threshold h
Å ai ( e ) 0 e and a process V̂ n ,1 as the process VV n ,1 corresponding to the threshold h
Å ai ( e ) / e, and check that they represent a lower and upper bound, respectively,
for V n ,1 .
Since on the interval [ t nk ( e, i), z nk ( e, i)), the process V n stays in the strip (ai ( e ) 0 e
/ r( e ), ai ( e ) / e 0 r( e )), the process V n ,1 up-crosses the level ai ( e ) 0 e in every interval
[ k nj01 , k nj ] belonging to [ t nk ( e, i) Ú T, z nk ( e, i) Ú T], so the construction above yields
Ṽ
n,1
t
Å
H
n,1
n ,
V t0
gI nj01/kj01
n
t √ [ gI j01
, bH jn ),
1 ° j ° q in,1
,k ,
V n,1
t0bH nj/c nj ,
t √ [ bH jn , gI jn ),
1 ° j ° q in,1
,k ,
where, in analogy with the proof of Theorem 3.1, n b˜ nj denotes the jth time a busy period
for Ṽ n ,1 starts and n g̃ nj denotes the jth time when the queue empties. By the fact that f is
nonnegative, we then get
*
q̃ n
(4.5)
f (V˜ n,1
t )dt °
0
*
z nk ( e,i ) ÚT
t kn ( e,i ) ÚT
f (V tn,1 )dt,
where q˜ n Å g̃ nq n,1
.
i,k
Similarly, since V n does not exceed ai ( e ) / e on [ t nk ( e, i) Ú T, z nk ( e, i) Ú T], the
construction above implies that
V n,1
Å
t
H
n ,
VP n,1
t0 k nj01/gP j01
t √ [ k nj01 , w jn ),
1 ° j ° q in,1
,k ,
n,1
VP t0
w nj/fP nj ,
t √ [ w jn , k jn ),
1 ° j ° q in,1
,k ,
where fP nj is defined as f nj above corresponding to h Å ai ( e ) / e and n gP nj is the jth time
when the queue empties. Again, since f is nonnegative,
*
z nk ( e,i ) ÚT
*
k 0nÚT
n,1
t
f (V )dt °
t nk ( e,i ) ÚT
t nk ( e,i ) ÚT
(4.6)
*
f (V n,1
t )dt
z kn ( e,i ) ÚT
/
k nq n,1
i,kÚT
f (V n,1
t )dt /
*
qO n
0
f (VP n,1
t )dt,
where qP n Å gP nq n,1
.
i,k
Thus, recalling that \ f \ Å supx f ( x), by (4.5) and (4.6) we have the bounds
*
f (V˜ n,1
t )dt °
*
°
*
q̃ n
0
z nk ( e,i ) ÚT
t nk ( e,i ) ÚT
qP n
(4.7)
0
f (V tn,1 )dt
f (VP tn,1 )dt / [ k 0n Ú T 0 t nk ( e, i) Ú T] \ f \
/ [ z kn ( e, i) Ú T 0 k nq n,1
Ú T] \ f \.
i,k
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
We next apply Theorem 3.1 to Ṽ n ,1 and V̂ n ,1 to get the asymptotics of the bounds on the
right and on the left.
Define
(4.8)
n
n
n
q̃ n,1
i ,k Å min( j : g̃ j/1 ú z k ( e, i) Ú T 0 t k ( e, i) Ú T ),
(4.9)
P nj/1 ú z nk ( e, i) Ú T 0 t nk ( e, i) Ú T ).
qP n,1
i ,k Å min( j : g
Obviously,
n,1
˜ n,1
qP n,1
i ,k ° q i ,k ° q i ,k .
(4.10)
Let U kn ( e, i) and V kn ( e, i) denote respectively the lower bound in (4.7) with q˜ n ( Åg̃ qn n,1
)
i,k
n
n
n
n
P
n,1
n,1
changed to ṽ Å g̃ qP i,k , and the upper bound in (4.7) with q ( ÅgP q i,k ) changed to vP n
Å gP q̃n n,1
. By (4.10),
i,k
*
z nk ( e,i ) ÚT
n
k
U ( e, i) °
(4.11)
t nk ( e,i ) ÚT
n
f (V n,1
t )dt ° V k ( e, i).
We now show that, as n r ` ,
d
U nk ( e, i) r Uk ( e, i),
(4.12)
d
V nk ( e, i) r Vk ( e, i),
k ¢ 1, 0 ° i ° N,
where
Uk ( e, i) Å
(4.13)
Vk ( e, i) Å
ai ( e ) 0 e
( zk ( e, i) Ú T 0 tk ( e, i) Ú T )
ai ( e ) / e
ai ( e ) / e
( zk ( e, i) Ú T 0 tk ( e, i) Ú T )
ai ( e ) 0 e
* f (u(a ( e ) 0 e ))du,
1
i
0
* f (u(a ( e ) / e ))du,
1
i
0
Let
n
q̃ n,1
i ,k (t) Å min( j ¢ 0 : g̃ j/1 ú t)
(4.14)
and
n
P j/1
ú t).
qP n,1
i ,k (t) Å min( j ¢ 0 : g
(4.15)
Note that
(4.16)
n
˜ n,1 n
q˜ n,1
i ,k Å q i ,k ( z k ( e, i) Ú T 0 t k ( e, i) Ú T ),
(4.17)
n
P n,1
qP in,1
,k Å q i ,k ( e, i) Ú T 0 t k ( e, i) Ú T ).
In the course of proving Theorem 3.1 we established (3.6). Since Ṽn ,1 and V̂ n ,1 meet
the conditions of Theorem 3.1 and the g̃ nj and gP nj are analogues of the g nj from (3.5), we
can write for these processes, in analogy with (3.6),
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S
P
g̃ nq_
 nt 
r t(ai ( e ) 0 e ) m1
D
1
1
/
,
l1 m1 0 l1
S
P
gP nq_
 nt 
r t(ai ( e ) / e ) m1
By Lemma 2.1 in Coffman et al. (1995) and (4.14), (4.15),
q˜ n,1
t
i ,k (t) P
r
q_
m
(a
(
e
) 0 e)
1
i
n
(4.18)
qP n,1
t
i ,k (t) P
r
q_
m1 (ai ( e ) / e )
n
S
S
1
1
/
l1 m1 0 l1
1
1
/
l1 m1 0 l1
D
D
279
D
1
1
/
.
l1 m1 0 l1
01
,
01
.
Lemma 2.2 in Coffman et al. (1995) then yields
P
n,1( t ) r
g̃ nq˜ i,k
(4.19)
ai ( e ) 0 e
t,
ai ( e ) / e
P
gP nq˜ n,1
r
i,k ( t )
ai ( e ) / e
t.
ai ( e ) 0 e
By Theorem 3.1 applied to Ṽ n ,1 and V̂ n ,1 and (4.19),
*
ai ( e ) 0 e
f (V˜ n,1
t
s )ds r
ai ( e ) / e
* f (u(a ( e ) 0 e ))du,
*
f (VP n,1
s )ds r
ai ( e ) / e
t
ai ( e ) 0 e
* f (u(a ( e ) / e ))du.
n,1
gP nq˜ i,k
(t)
0
(4.20)
gP nq˜ n,1
i,k (t)
0
P
P
1
i
0
1
i
0
In view of (4.16), Lemma 2.2 in Coffman et al. (1995) shows that (4.4) and (4.20)
imply
0
ai ( e ) 0 e
( zk ( e, i) Ú T 0 tk ( e, i) Ú T )
ai ( e ) / e
*
* f (u(a ( e ) 0 e ))du,
*
ai ( e ) / e
f (VP n,1
( zk ( e, i) Ú T 0 tk ( e, i) Ú T )
t )dt r
ai ( e ) 0 e
* f (u(a ( e ) / e ))du.
ṽ n
d
f (V˜ n,1
t )dt r
vP n
0
d
1
i
0
1
i
0
P
P
Since Ék n0 Ú T 0 t nk ( e, i) Ú T É r 0 and Éz nk ( e, i) Ú T 0 k nq i,kn,1 Ú T É r 0 obviously hold,
(4.12) is proved. Moreover, the same argument shows that
d
˜ (Uk ( e, i))k¢1,0°i°N ),
(V n , (U nk ( e, i))k¢1,0°i°N ) r (V,
(4.21)
d
˜ (Vk ( e, i))k¢1,0°i°N ).
(V n , (V nk ( e, i))k¢1,0°i°N ) r (V,
Next, defining
`
(4.22)
N
`
N
U n ( e ) Å ∑ ∑ U kn ( e, i),
V n ( e ) Å ∑ ∑ V kn ( e, i),
kÅ1 iÅ0
kÅ1 iÅ0
we need to prove that
(4.23)
d
˜ U( e )),
(V n , U n ( e )) r (V,
d
˜ V( e )),
(V n , V n ( e )) r (V,
where
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
`
`
N
U( e ) Å ∑ ∑ Uk ( e, i),
(4.24)
N
V( e ) Å ∑ ∑ Vk ( e, i).
kÅ1 iÅ0
kÅ1 iÅ0
We prove the first convergence result in (4.23); the proof of the second uses the same
reasoning.
Since U nk ( e, i) Å 0 if t nk ( e, i) ¢ T, we have by (4.3) and (4.4),
SZ
lim lim P
Mr` nr`
M
Z D
N
∑ ∑ U nk ( e, i) 0 U n ( e ) ú 0
kÅ1 iÅ0
° lim lim P( min z nM ( e, i) Ú (T / 1) õ T )
(4.25)
Mr` nr`
0°i°N
° lim P( min zM ( e, i) Ú (T / 1) ° T ) Å 0.
Mr`
0°i°N
Analogously,
M
N
P
∑ ∑ Uk ( e, i) r U( e )
(4.26)
(M r ` ).
kÅ1 iÅ0
Next, by (4.21) and the continuous mapping theorem, we have
S
M
D S
N
d
M
N
˜ ∑ ∑ Uk ( e, i)
V n , ∑ ∑ U kn ( e, i) r V,
(4.27)
kÅ1 iÅ0
kÅ1 iÅ0
D
.
d
The convergence (V n , U n ( e )) r ( Ṽ, U( e )) then follows from (4.24) – (4.27) and Theorem 4.2 in (Billingsley (1968)).
Now by the definition of t nk ( e, i) and z nk ( e, i),
Z*
T
0
`
N
n
f (V n,1
t )·1( d ° V t ° K)dt 0 ∑ ∑
kÅ1 iÅ0
* [1( d 0 e ° V
* f (V
Z
T
n,1
t
)·1(t √ [ t kn ( e, i), z kn ( e, i)))dt
0
T
° \f \
n
t
° d ) / 1(K ° V tn ° K / e )]dt
0
so by (4.11), we obtain from (4.22)
* [1( d 0 e ° V
T
Un(e) 0 \ f \
n
t
° d ) / 1(K ° V tn ° K / e )]dt
0
* f (V
T
(4.28)
°
n,1
t
)·1( d ° V tn ° K)dt
0
* [1( d 0 e ° V
T
° Vn(e) / \ f \
n
t
° d ) / 1(K ° V tn ° K / e )]dt.
0
Therefore, if we prove that as e r 0
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* S * f (uV˜ )duD 1( d ° V˜ ° K)dt,
T
d
U( e ) r
1
t
0
t
0
(4.29)
* S * f (uV˜ )duD 1( d ° V˜ ° K)dt,
T
d
V( e ) r
1
t
0
t
0
then by applying Lemma 2.3 in Coffman et al. (1995) to (4.29) and taking into account
(4.23), (4.2), we will then obtain (4.1). As before, we prove only the first of the results
in (4.29); the proof to the second is similar.
In fact, we prove convergence with probability 1. The argument is almost identical to
that in Coffman et al. (1995), but we give it here since it is used once again below. Since
ai ( e ) ú d, we have from (4.13) and (4.24)
Z
`
* f (u(a ( e ) 0 e ))du[ z ( e, i) Ú T 0 t ( e, i) Ú T] Z ° 2de \ f \T.
N
1
U( e ) 0 ∑ ∑
i
k
k
0
kÅ1 iÅ0
This tends to 0 as e r 0, so we prove that
`
N
lim ∑ ∑ [ zk ( e, i) Ú T 0 tk ( e, i) Ú T]
er0 kÅ1 iÅ0
* f (u(a ( e ) 0 e ))du
1
i
0
* S * f (uV˜ )duD 1( d ° V˜ ° K)dt.
(4.30)
T
Å
1
t
0
t
0
We can write
`
* f (u(a ( e ) 0 e ))du
N
1
∑ ∑ [ zk ( e, i) Ú T 0 tk ( e, i) Ú T]
i
0
kÅ1 iÅ0
`
(4.31)
Å
N
∑ ∑
kÅ1 iÅ0
* S * f (u(a ( e ) 0 e ))duD·1( t ( e, i) ° t õ z ( e, i))dt
T
1
i
0
k
k
0
å Ce .
Note that if x, y ú d /2, É x 0 yÉ õ 2e, then
Z*
*
1
1
f (ux)du 0
0
°
0
Z
1 1
0
x y
Z*
x
0
Z Z*
1
x
f (uy)du Å
f (u)du /
1
y
Z*
y
x
x
f (u)du 0
0
Z
f (u)du °
1
y
* f (u)duZ
y
0
8e
\ f \.
d
Since Ṽt √ [ai ( e ) 0 e, ai ( e ) / e], for t √ [ tk ( e, i), zk ( e, i)), we then have
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
Z*
1
f (u(ai ( e ) 0 e )du·1(t √ [ tk ( e, i), zk ( e, i)))
0
* f (uV˜ )du·1(t √ [ t ( e, i), z ( e, i))) Z
1
0
t
k
k
0
°
8e
\ f \·1(t √ [ tk ( e, i), zk ( e, i))),
d
so by (4.31)
Z
`
* S * f (uV˜ )duD·1(t √ [ t ( e, i), z ( e, i)))dtZ ° 8de \ f \T,
N
T
Ce 0 ∑ ∑
1
t
0
kÅ1 iÅ0
k
k
0
whence
Z * S*
T
0
D
1
Ce 0
Z
f (uV˜ t )du ·1( d ° V˜ t ° K)dt
0
* [1( d ° V˜ ° d 0 e ) / 1(K ° V˜ ° K / e )]dt / 8de \ f \T.
T
° \f \
t
t
0
Since by (4.2) the right-hand side of this inequality tends to 0 as e r 0, we have proved
(4.30). This completes the proof of the first assertion of Theorem 4.1 for bounded nonnegative f ( x ). The general case is handled via a localization argument as in the proof of
Theorem 2.1 in Coffman et al. (1995).
5. Tightness results. The purpose of this section is to prove several results on the
tightness of some processes closely related to the normalized and time-scaled unfinishedwork process V n defined in (2.8). We start, however, with a simple fact.
Introduce
B n,l
t Å
(5.1)
n
S n,l
nt 0 r l nt
,
q_
n
B nt Å
l Å 1, 2,
S nnt 0 nt
,
q_
n
n
where S n,l
and S nt were defined in (2.11), and let B n , l Å (B n,l
t
t , t ¢ 0), l Å 1, 2, B
n
Å (B t , t ¢ 0).
LEMMA 5.1.
As n r ` ,
q_
1  nt  n,l P
q_ ∑ s i r dlt,
n iÅ1
d
B n , l r ll1 / 2 slW l ,
1  nt  n,l P
∑ s i r dlt,
n iÅ1
l Å 1, 2,
A ntn,l P
r llt,
n
l Å 1, 2,
d
B n r ( sWt / ct, t ¢ 0),
where W l , l Å 1, 2, and W Å (Wt , t ¢ 0) are standard Brownian motions.
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283
PROOF . We give proofs of the first and second
convergence
results in the first line.
q_
q_
n,l
n,l 2
n,l
n
0,
Var
s
°
E(s
)
1(s
ú
e
n
)
/
e
nd
,
it
follows
by (2.2) and (2.6)
Since, for e ú
1
1
1
l
q_
that Var s n,l
/
n
r
0
as
n
r
`
.
Hence,
1
q_
P
1  nt  n,l
n
q_ ∑ (s i 0 d l ) r 0,
n iÅ1
P
1  nt  n,l
∑ (s i 0 d nl ) r 0,
n iÅ1
and both convergences follow by (2.2).
P
 nt 
j n,l
r ll01 t, which is
The third convergence follows from the convergence 1/n ( iÅ1
i
proved similarly, and properties of the first-passage-time map (Whitt 1980; see also Coffman et al. 1995, Lemma 2.1). The claimed convergences in distribution hold by the
h
Donsker-Prohorov invariance principle and (2.3).
Let b nt denote the indicator of the event that the server is switching over at nt, i.e.,
b nt Å 1 0 a nnt . Then definitions (2.8), (2.11), (5.1) and Equation (2.10) imply that V nt
satisfies
t
t
q_
q_
(5.2)
V nt Å V 0n / B tn / n
1(V sn Å 0)ds / n
1(V sn ú 0) b sn ds.
*
*
0
0
We study properties of the processes on the right. For e ú 0, we define the processes K n ,e
Å (K n,e
t , t ¢ 0) by
t
q_
(5.3)
K n,e
1(V sn ú e ) b sn ds.
t Å n
*
0
Though the K n ,e have continuous paths, we still consider them as random elements of
D[0, ` ).
Recall that a sequence of processes {X n , n ¢ 1} in D[0, ` ) is called C-tight if it is
tight and all weak limit points of the sequence of their laws are laws of continuous
processes (Jacod and Shiryaev (1987), VI.3.25). Below, we repeatedly use the fact that
{X n , n ¢ 1} is C-tight if and only if, for all T ú 0 and h ú 0,
lim lim P(É X n0 É ú H) Å 0,
Hr` nr`
lim lim P(
dr0 nr`
sup
É X nt 0 X ns É ú h ) Å 0
s,t°T,És0tÉ°d
(this follows, e.g., from Jacod and Shiryaev (1987), VI.3.26).
Another technical tool used below is the concept of strong majorization (Jacod and
Shiryaev (1987), VI.3.34): Say that a process X Å (Xt , t ¢ 0) strongly majorizes a
process Y Å (Yt , t ¢ 0) if the process X 0 Y Å (Xt 0 Yt , t ¢ 0) is nondecreasing. If a
sequence {X n , n ¢ 1} of processes is C-tight and each X n strongly majorizes a process
Y n , where X n and Y n are both nondecreasing and start at 0, then the sequence {Y n , n
¢ 1} is C-tight (Jacod and Shiryaev (1987), VI.3.35).
LEMMA 5.2.
every e ú 0.
The sequence {K n ,e , n ¢ 1}, where K n ,e Å (K n,e
t , t ¢ 0), is C-tight for
n,1
n,2
n,2
PROOF . Denote by [u n,1
i , £ i ] and [u i , £ i ], i ¢ 1, the respective successive switchover periods (i.e., times during which the server is switching) from the first queue to the
second and from the second queue back to the first. Let q n,l
t be the number of switchovers
from queue l started in [0, nt]. By (5.3),
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
n,l
1 2 qt
_ ∑ ∑
Å
K n,e
q
t
n lÅ1 iÅ1
(5.4)
*
£ n,l
i Únt
u in,lÚnt
q_
1(U ns ú e n )ds.
So, if we define
n,l
1 2 qt
˘ n,e
_ ∑ ∑
Å
K
q
t
n lÅ1 iÅ1
(5.5)
*
£ n,l
i
u n,l
i
q_
1(U ns ú e n )ds,
then, obviously,
(5.6)
1
n,l
n,l
˘ n,e
supÉK n,e
s 0 Ks É ° q_ max max ( £ i 0 u i ).
n,l
s°t
lÅ1,2
1°i°q
t
n
Note that since
S
P
q n,l
t/1
t
ú
n
dl
D S
°P
 n ( t/1 ) /dl 
∑
D
s in,l ° nt ,
iÅ1
Lemma 5.1 implies that
(5.7)
S
lim P
nr`
q n,l
t/1
t
ú
n
dl
D
Å 0.
Recalling also that
n,l
n,l
£ n,l
i 0 ui Å si ,
(5.8)
we get by Lemma 3.1 and (2.6) that the right-hand side of (5.6) tends in probability to
0 as n r ` . Thus the C-tightness of {K n ,e , n ¢ 1} will follow from the C-tightness of
{K̆ n ,e , n ¢ 1}.
Define
n,l
q_
1 qt
n
˜ n,e,l
(5.9)
K
Å q_ ∑ s n,l
ú
e
n ),
l Å 1, 2.
t
i ·1(U £ n,l
i
n iÅ1
n,l
n ,e
is strongly majorized
Since U ns cannot increase on [u n,l
i , £ i ], we have from (5.5) that K̆
n ,e
n ,e,1
n ,e,2
n ,e, l
n,e,l
˜
by K̃ Å K̃
/ K̃
, where K̃
Å (K t , t ¢ 0), l Å 1, 2. Therefore, it is enough
to prove that {K̃ n ,e,1 , n ¢ 1} and {K̃ n ,e,2 , n ¢ 1} are each C-tight. By symmetry, we need
only prove that {K̃ n ,e,1 , n ¢ 1} is C-tight.
Let
n,1
q_
1 qt
n
(5.10)
KV n,e,1
Å q_ ∑ s n,1
ú
e
n /2),
t
i ·1(U u n,1
i
n iÅ1
n,1
q_
q_
1 q t n,1
n,e,1
P
_
(5.11)
K t Å q ∑ s i ·1(U £nin,1 ú e n , U un in,1 ° e n /2).
n iÅ1
, t ¢ 0) and
By (5.9), K̃ n ,e,1 is strongly majorized by KV n ,e,1 / K̂ n ,e,1 , where KV n ,e,1 Å (KV n,e,1
t
K̂ n ,e,1 Å (KP n,e,1
, t ¢ 0). We prove that {KV n ,e,1 , n ¢ 1} is C-tight and that KP tn,e,1 tends in
t
probability to 0 uniformly over finite intervals as n r ` . This will conclude the proof of
the lemma.
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285
n,1
We begin with the property of K̂ n ,e,1 . Since there is no service on [u n,1
i , £ i ], we have
by (5.11) and (2.11),
n,1
q_
1 q t n,1
PK n,e,1
_
° q ∑ s i ·1(S £nin,1 0 S un n,1
ú e n /2)
t
i
n iÅ1
n,1
q_
1 qt
n,1
n,1
n,1
n,1
° q_ ∑ s n,1
·
1(S
0
S
ú
e
n /4)
i
£i
ui
n iÅ1
n,1
q_
1 q t n,1
n,2
_
/ q ∑ s i ·1(S n,2
£ in,1 0 S uin,1 ú e n /4),
n iÅ1
so by (5.8), for d ú 0, since u n,1
q tn,1 ° nt,
(5.12)
S
D S
D
q n,1
t/1
1
t
P(KP n,e,1
ú 0) ° P
ú
/ P q_
sup
s n,1
úd
t
i
n
d1
1°i°

n
(
t/1)
/d

1
n
q_
q_
n,1
n,2
/ P( sup É S n,1
sup É S n,2
£ 0 S u É ú e n /4) / P(
£ 0 S u É ú e n /4).
u°nt q_
0°v0u°d n
u°nt q_
0°v0u°d n
The first term on the right of (5.12) goes to 0 as n r ` by (5.7). The second term tends
to 0 as n r ` by Lemma 3.1 and (2.6).
Next,
P(
q_
n,1
É S n,1
£ 0 S u É ú e n /4)
sup
u°nt q_
0°£0u°d n
S
Å P
(5.13)
S
S
Z
n,1
n,1
S n£
0 r 1n n £ S nu
0 r 1n nu
e
0
ú 0 r 1n d
q_
q_
4
n
n
q_ u°t
0° n ( £0u ) °d
° P
sup
u°t,Éu0£É°g
ÅP
Z D
Z
n
n,1
q_
S n,1
S nu
0 r n1 nu
e
n£ 0 r 1 n £
0
/ r 1n n ( £ 0 u) ú
q_
q_
4
n
n
sup
n,1
É B n,1
£ 0 Bu É ú
sup
D
Z
u°t,Éu0£É°g
D
e
0 r n1 d ,
4
where g ú 0 is arbitrary and n is large enough. Since by Lemma 5.1, B n ,1 converges in
distribution to l11 / 2 s1W 1 , and since the functional X r sup É X£ 0 XuÉ, X √ D[0, ` ),
u°t,Éu0£É°g
is continuous almost everywhere with respect to the Wiener measure (Liptser and Shiryaev
(1989)), we conclude from (5.13) and (2.3) that
lim P(
sup
nr`
u°nt q_
0°£0u°d n
S
°P
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q_
n,1
É S n,1
0
S
É
ú
e
n /4)
£
u
sup
/ 2 01
ÉW 1£ 0 W u1 É ¢ l01
s1
1
u°t,Éu0£É°g
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e
0 r1d
4
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
which goes to 0 as g r 0, if d õ e /4r1 , by the continuity of Brownian motion. We have
thus proved that the third term on the right of (5.12) tends to 0 as n r ` if d is small
enough. A similar argument applies to the last term on the right-hand side of (5.12).
Therefore, since K̂ n ,e,1 is nondecreasing,
lim P(sup KP n,e,1
ú 0) Å 0,
s
nr`
s°t
as required.
We now prove that {KV n ,e,1 , n ¢ 1} is C-tight. Call a switchover from queue 1 to queue
2 sound if at the time when it starts, the total unfinished
work (which at that moment is
q_
the unfinished work at queue 2) is greater than e n /2. Let qV n,1
be the number of sound
t
switchovers started in [0, nt]. By (5.10),
V n,1
1 qt
Å q_ ∑ sV n,1
KV n,e,1
t
i ,
n iÅ1
(5.14)
where sV n,1
is the duration of the ith sound switchover. Note that the soundness of a
i
switchover is determined at its beginning, so the sV n,1
i , i ¢ 1, are i.i.d. and distributed as
the s n,1
i , i ¢ 1.
We have by (5.14), for t ú 0, d ú 0, h ú 0, g ú 0 and L ú 0,
P(
ÉKV n,e,1
0 KV n,e,1
É ú h)
£
u
sup
u,£°t,Éu0£É°d
q_
° P(qV n,1
t ú L n ) / P(
q_
V n,1
ÉqV n,1
£ 0 qu É ú g n )
sup
£0 d°u°£°t
(5.15)
S
/P
sup
£0 g°u°£°L
Z
Z D
q_
1 £ n
úh .
q_
∑_ sV n,1
i
n iÅ  u n  /1
q
V n,1
Now, if qV n,1
£ 0 q u Å m, then the amount of work
q_ executed by the server at queue
q_2 in
the interval [nu,q_
n £ ] is no less than (m 0 1) e n /2 which takes time (m 0 1)
e n /2.
q_
0 qV n,1
Hence (m 0 1) e n /2 ° n( £ 0 u) which leads to the estimate qV n,1
£
u ° (2 n / e )( £
0 u) / 1, so that, for all n large enough,
q_
3
n
V n,1
sup ÉqV n,1
d;
£ 0 qu É °
e
£0 d°u°£°t
Taking in (5.15) L Å 3et and g Å 3e d, we get
lim P(
nr`
sup
ÉKV
u,£°t,Éu0£É°d
n,e,1
£
0 KV
n,e,1
u
S
É ú h ) ° lim P
nr`
q_
3
n
qV tn,1 °
t.
e
sup
£03d / e°u°£°3t / e
Z
q_
q
where the latter limit, by Lemma 5.1, is zero if (3d / e )d1 õ h. Therefore,
lim lim P(
dr0 nr`
/ 3904 0014 Mp
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sup
ÉKV n,e,1
0 KV n,e,1
É ú h ) Å 0,
£
u
u,£°t,Éu0£É°d
Wednesday May 06 03:12 PM
Z D
1 £ n 
úh ,
q_
∑_ sV n,1
i
n iÅ  u n  /1
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POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
which, since KV n,e,1
Å 0, proves the C-tightness of {KV £n,e,1 , n ¢ 1}. The lemma is
0
proved.
h
We next prove that the two rightmost processes in (5.2) are asymptotically bounded
in probability.
LEMMA 5.3.
We have
S *
q_
(5.16)
lim lim P
Ar` nr`
S *
lim lim P
Ar` nr`
PROOF .
1(V ns Å 0)ds ú A Å 0,
0
q_
(5.17)
D
t
n
D
t
1(V ns ú 0) b ns ds ú A Å 0.
n
0
Let
w nt Å V nt 0 V 0n 0 B tn 0 K tn,1 .
(5.18)
By (5.3), (5.2) and the inequality 0 ° b nt ° 1, we have for 0 õ s õ t,
q_
w nt 0 w ns Å n
* 1(V
t
n
u
* 1(0 õ V
q_
Å 0)du / n
t
s
n
u
q_
° 1) b nu du ° n
s
* 1(V
t
n
u
° 1)du,
s
so, since V nu ¢ w nu / B nu by (5.18),
q_
w nt 0 w ns ° n
* 1( w
t
n
u
° 1 0 B nu )du.
s
Therefore, by Lemma 1 in Coffman, Puhalskii, and Reiman (1991)
w nt ° sup(1 0 B ns ) Û 0.
s°t
Since the sequence {B n , n ¢ 1} is C-tight by Lemma 5.1, we conclude that
lim lim P( w nt ú A) Å 0.
Ar` nr`
Since by (5.2) and (5.18)
q_
w nt / K n,1
Å n
t
*
t
q_
1(V sn Å 0)ds / n
0
* 1(V
t
n
s
ú 0) b sn ds,
0
an application of Lemma 5.2 completes the proof.
h
We are in need of two more technical lemmas. Introduce the processes
n,l
(5.19)
1 A nt /1 n,l
n n,l
M n,l
t Å q_ ∑ ( h i 0 r l j i ),
n iÅ1
l Å 1, 2,
M nt Å M n,1
/ M n,2
t
t ,
n,l
, l Å 1, 2.
and recall that t n,l
i , i Å 1, 2, ··· denote arrival times for A
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
LEMMA 5.4. Define the filtration F n Å (F nt , t ¢ 0) by F nt Å F n,1
Û F tn,2 Û s(s in,l , i
t
n
n,l
n,l
n,l
n,l
Å 1, 2, . . . , l Å 1, 2) Û s(V 0 ) Û N, where F t Å G A n,l
, G i Å s( h j , j n,l
j , 1 ° j ° i),
nt /1
n
l Å 1, 2, and N is the family of P-null sets. Then F is well defined, the t n,l
i /n, i Å 1, 2,
n
. . . , l Å 1, 2, are F n-stopping times, the processes (A n,l
nt , t ¢ 0), l Å 1, 2, are F n
n
n
predictable and M Å (M t , t ¢ 0) is an F -locally square-integrable martingale with the
predictable quadratic-variation process
» M n …t Å
1
[( s 1n ) 2 A ntn,1 / ( s 2n ) 2 A ntn,2 ].
n
PROOF . The proof is almost the same as the proof of Lemma 2 in Coffman et al.
(1991). In particular, the martingale property of M n and the formula for its predictable
quadratic-variation process is deduced from the fact that the processes ( ( kiÅ1 ( h in,l
0 r nl j n,l
i ), k ¢ 0), l Å 1, 2, are locally square-integrable martingales which have the
predictable quadratic-variation processes (( s nl ) 2k, k ¢ 0) relative to the respective flows
(G n,l
h
k , k ¢ 0).
Note that the processes B n , ( b nt , t ¢ 0) and V n are F n-adapted.
Introduce
(5.20)
r nl
e n,l
t Å q_
n
F∑
A n,l
nt /1
iÅ1
G
1
j n,l
0 q_ h An,ln,l
,
i 0 nt
nt /1
n
l Å 1, 2,
e nt Å e n,1
/ e tn,2 .
t
Let DM ns denote the jump of M n at s.
LEMMA 5.5.
Under the hypotheses of Theorem 2.3, for t ú 0,
P
» M n …t r s 2t,
P
P
∑ ( DM sn ) 2 r s 2t,
supÉe ns É r 0,
s°t
0õs°t
as n r ` .
PROOF . The first convergence follows by the expression for » M n …t in Lemma 5.4,
Lemma 5.1, (2.1) and (2.7). For the second, note that since M n is a process of locally
bounded variation by (5.19), it is a purely discontinuous local martingale (Jacod and
Shiryaev (1987), I.4.14; Liptser and Shiryaev (1989), I.7), so its quadratic-variation
process ([M n , M n ]t , t ¢ 0) is the sum of the squares of jumps: [M n , M n ]t Å (s°t
( DM ns ) 2 (Jacod and Shiryaev (1987), I.4.52; Liptser and Shiryaev (1989), I.8). By
Lemma 5.5.5 in Liptser and
Shiryaev (1989),
(2.4), (2.5) and Lemma 5.1 imply that the
P
P
convergences [M n , M n ]t r s 2t and » M n …t r s 2t are equivalent, so the second convergence
of the lemma is a consequence of the first. The third convergence results from the inequalities
n,l/1
A nt
n,l
0 ° ∑ j n,l
,
i 0 nt ° j A n,l
nt /1
iÅ1
conditions (2.4) and (2.5) and Lemmas 3.1 and 5.1.
Let
h
VV nt Å V nt 0 e nt .
(5.21)
Since V n is F n-adapted and e nt is F nt -measurable by (5.20), VV n Å (VV nt , t ¢ 0) is F nadapted. By (5.1), (5.2), (5.19) and (5.20), we get the representation
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q_
q_
VV nt Å VV 0n / n ( r n 0 1)t / n
(5.22)
q_
/ n
* 1(V
* 1 (V
t
n
s
Å 0)ds
0
t
n
s
ú 0) b ns ds / (M nt 0 M 0n ).
0
Now squaring in (5.22), we have by Ito’s formula (Theorem 2.3.1 in Liptser and Shiryaev
(1989)) that
q_
(VV nt ) 2 Å (VV n0 ) 2 / 2 n ( r n 0 1)
(5.23)
q_
/2 n
*
*
t
0
q_
VV ns ds / 2 n
t
0
* VV
* VV 1 (V
t
n
s
t
VV ns 1(V ns ú 0) b ns ds / 2
0
n
s
Å 0)ds
0
n
s0
dM ns / ∑ ( DM ns ) 2 ,
0õs°t
where VV ns0 denotes the left-hand limit of VV n at s.
LEMMA 5.6.
The sequences {VV n , n ¢ 1} and {V n , n ¢ 1} are C-tight.
PdROOF . By (5.16), (5.17), the C-tightness of {B n , n ¢ 1}, and the convergence
V r V0 , the right-hand side of (5.2) is asymptotically bounded in probability, i.e.,
n
0
lim lim P(supÉV ns É ú A) Å 0,
(5.24)
Ar` nr`
t ú 0.
s°t
Then (5.21) and Lemma 5.5 yield
lim lim P(supÉVV ns É ú A) Å 0,
(5.25)
Ar` nr`
t ú 0.
s°t
We now check that, for any T ú 0 and h ú 0, we have that
(5.26)
lim lim
sup P(supÉ(VV nt/t ) 2 0 (VV nt ) 2É ú h ) Å 0,
dr0 nr` t√ ST ( F n )
t°d
where ST ( F n ) is the set of all F n-stopping times t not greater than T.
Since the processes
S*
t
0
n
VV s0
dM ns , t ¢ 0
D
and
S∑
( DM ns ) 2 0 » M n …t ,
t¢0
0õs°t
D
are F n-local martingales (Liptser and Shiryaev (1989), Ch. 1, §8, Ch. 2, §2), the LenglartRebolledo inequality (Liptser and Shiryaev (1989), Theorem 1.9.3) yields, in view of
(5.23), for e ú 0,
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
e
P(supÉ(VV nt/t ) 2 0 (VV nt ) 2É ú h ) °
h
t°d
S
q_
/ P 2 n Ér n 0 1É
(5.27)
*
q_
ÉVV un É du / 2 n
t/d
t
t/d
ÉVV un É1 (V un Å 0)du
t
D
*
q_
/2 n
*
t/d
ÉVV nu É1(V nu ú 0) b nu du / » M n …t/d 0 » M n …t ú e .
t
q_
By (5.25) and the assumed limit n ( r n 0 1) r c, we have
S
q_
lim lim P 2 n Ér n 0 1É sup
(5.28)
dr0 nr`
És0tÉ°d
s°T
* ÉVV É du ú 4e D Å 0.
t
n
u
s
By (5.21), we see that ÉVV ns É1 (V ns Å 0) Å Ée ns É1 (V ns Å 0), so (5.16) and Lemma 5.5
yield
q_
2 n
(5.29)
* ÉVV É1 (V
t
n
s
n
s
P
Å 0)ds r 0(n r ` ),
t ú 0.
0
Next, for e* ú 0, 0 õ s õ t, we again use (5.21) and obtain
q_
2 n
* ÉVV É1(V
t
n
u
n
u
ú 0) b nu du
s
(5.30)
q_
° 2 n supÉe nu É
u°t
* 1(V
t
n
u
ú 0) b nu du
s
q_
/ 2 n supÉV nu É
u°t
*
t
q_
1(V nu ú e* ) b nu du / 2 n e*
s
* 1(V
t
n
u
ú 0) b nu du.
s
The first term on the right tends in probability to 0 as n r ` by Lemma 5.5 and (5.17).
The third term tends in probability to 0 as n r ` and then e* r 0 by (5.17). Finally, by
(5.3), Lemma 5.2 and (5.24), we have for g ú 0,
S
q_
lim lim P 2 n sup ÉV nu É sup
dr0 nr`
u°T
És0tÉõd
s°T
* 1(V
D
t
n
u
ú e* ) b nu du ú g Å 0.
s
Thus, by (5.30),
(5.31)
S
q_
lim lim P 2 n sup
dr0 nr`
És0tÉ°d
s°T
* ÉVV É·1(V
t
n
u
n
u
ú 0) b nu du ú
s
e
4
D
Å 0.
Lemma 5.5 easily implies that
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(5.32)
S
sup É» M n …t 0 » M n …sÉ ú
lim lim P
dr0 nr`
És0tÉõd
s°T
e
4
D
Å 0.
Applying (5.28), (5.29), (5.31) and (5.32) to (5.27) shows that (5.26) holds.
Now, by Aldous’ condition (see, e.g., Liptser and Shiryaev (1989), Theorem 6.3.1),
(5.25) and (5.26) imply that the sequence {(VV n ) 2 , n ¢ 1} and hence {VV n , n ¢ 1} is
tight for the Skorohod topology. By Proposition VI.3.26 in Jacod and Shiryaev (1987),
it remains to prove that
P
supÉDVV nt É r 0,
T ú 0.
t°T
In view of (5.19),
supÉDVV nt É Å supÉDM nt É ° max
t°T
t°T
lÅ1,2
F
r nl
h n,l
i / q_
nT /1
n 1°i°A n,l
n
1
q_
max
max
1°i°A n,l
nT /1
j n,l
i
G
which tends to 0 in probability as n r ` by Lemmas 3.1 and 5.1 and (2.3), (2.4) and
(2.5). This proves that {VV n , n ¢ 1} is C-tight. The sequence {V n , n ¢ 1} is then C-tight
h
by Lemma 5.5 and (5.21).
By Prohorov’s theorem, Lemma 5.6 makes it certain dthat there exists a subsequence
{V n = , n * ¢ 1} and a continuous process Ṽ such that V n = r Ṽ. The next two lemmas deal
with implications of this fact.
LEMMA 5.7.
We have , for h ú 0,
S*
er0 nr`
D
t
1(V ns õ e )ds ú h Å 0.
lim lim P
0
In particular, if the law of a process Ṽ Å (Ṽt , t ¢ 0) is an accumulation point of the
laws of {V n , n ¢ 1}, then
* 1 (V˜ Å 0)ds Å 0
t
a.s.
s
0
PROOF .
d
Since V n = r Ṽ for a subsequence (n * ), we have, for e ú 0 and h ú 0,
S*
lim P
nr`
D S*
t
0
D
t
1(V ns = õ e )ds ú h ¢ P
1(V˜ s õ e )ds ú h ,
0
and the second assertion of the lemma is a consequence of the first. To prove that, introduce the processes Z n Å (Z nt , t ¢ 0) by
q_
Z nt Å V n0 / B tn / n
* 1(V
t
n
s
ú 0) b ns ds,
t ¢ 0,
0
so that (5.2) is equivalent to
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
q_
V nt Å Z nt / n
* 1 (V
t
n
s
Å 0)ds.
0
q_
t
Since V nt is nonnegative and n *0 1 (V ns Å 0) ds increases only when V nt equals 0, we
n
n
conclude that V Å R (Z ), where R : D[0, ` ) r D[0, ` ) is Skorohod’s reflection map.
In the one-dimensional case it is well known to be equivalently defined by (for C[0, ` ),
the result is in Ikeda and Watanabe (1989), the result for D[0, ` ) is a special case of a
more general result in Chen and Mandelbaum (1991))
R(X )t Å Xt 0 inf Xs Ú 0,
(5.33)
t ¢ 0.
s°t
Now, if we define
Zd nt Å V 0n / B tn ,
(5.34)
Zd n Å (Zd tn , t ¢ 0),
t ¢ 0,
and introduce
Vd n Å R(Zd n ),
(5.35)
then the process Z n 0 Ž n is increasing and (5.33) implies that V nt ¢ Vd nt , t ¢ 0. Hence,
for e ú 0,
S*
D S*
t
(5.36)
0
D
t
1(V sn õ e )ds ú h ° P
P
0
1(Vd sn õ e )ds ú h .
Now, by (5.34) and Lemma 5.1, {Ž n , n ¢ 1} converges in distribution to the process Ž
Å (V0 / sWt / ct, t ¢ 0), where V0 and (Wt , t ¢ 0) are independent. By the continuity
d
of the reflection (Whitt (1980), Theorem 6.4), we then deduce that V̌ n r R( Ž ), i.e., by
(5.36),
S*
S*
D
t
lim limP
er0 nr`
1(V ns õ e )ds ú h ° lim limP
0
er0
0
S*
0
D S*
t
° lim P
er0 nr`
1(R(Zd )s ° e )ds ¢ h Å P
D
D
t
1(Vd ns õ e )ds ú h
t
0
1(R(Zd )s Å 0)ds ¢ h .
Since R(Ž ) is a reflected Brownian motion, theqlatter
probability equals 0.
h
_
The next lemma shows that, ‘‘on average,’’ n b nt behaves as d/V nt substantiating the
heuristic argument of §1.
LEMMA 5.8.
Under the conditions of Theorem 2.1, for T ú 0,
q_
lim n
nr`
* b V dt Å dT.
T
n
t
n
t
0
PROOF . We rely heavily on the argument in the proof of Theorem 4.1. The notation
in that proof is used here. Again let a continuous process Ṽ Å ( Ṽt , t ¢ 0) be an accu-
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mulation point of {V n , n ¢ 1}. By Lemma 5.7, the conditions of Theorem 4.1 hold, so
the results developed in the proof of Theorem 4.1 apply.
As in the proof of Theorem 4.1, it is enough to prove
* V b ·1( d ° V
q_
(5.37)
T
n
t
n
n
t
n
t
* 1( d ° V˜ ° K)dt
T
d
° K)ds r d
0
t
0
for any d and K, 0 õ d õ K that satisfy (4.2). To check this, note first that, for h ú 0,
S *
S *
q_
(5.38)
lim lim P
dr0 nr`
V nt b nt ·1(V nt ° d )dt ú h Å 0,
0
q_
(5.39)
D
D
T
n
lim lim P
Kr` nr`
T
n
V nt b nt ·1(V nt ¢ K)dt ú h Å 0.
0
Limit (5.38) follows from (5.17). For (5.39), write
S *
q_
P
D
T
n
V nt b nt ·1(V nt ¢ K)dt ú h ° P(sup V nt ¢ K)
0
t°T
and observe that the latter goes to 0 as n r ` and K r ` by the tightness of {V n , n
¢ 1}. Theorem 4.2 in Billingsley (1968) then implies the desired result by (5.38), (5.39)
and the fact that, by Lemma 5.7 and the continuity of Ṽ,
* 1(V˜ ° d )dt Å * 1 (V˜ Å 0)dt Å 0, lim * 1(V˜ ú K)dt Å 0,
T
lim
dr0
T
T
t
t
0
t
Kr`
0
P 0 a.s.
0
So, we next prove (5.37) assuming (4.2).
n
Recall that k n,1
j (i, k), j ¢ 0, are the successive times after n t k ( e, i), when the unfinished
n,1
work at queue 1 becomes equal to 0, and q i ,k is the number of cycles accumulation-service
for queue 1 in [ t nk ( e, i) Ú T, z nk ( e, i) Ú T]. We denote the switchover times starting after
n,1
n t nk ( e, i) by s n,1
0 (i, k), s 1 (i, k ), . . . . Obviously, they are independent and distributed
n,1
as s 1 . We introduce a similar notation for queue 2: n k n,2
j (i, k), j ¢ 0, denote the successive times after n t nk ( e, i), when the unfinished work at queue 2 becomes equal to 0,
n
q n,2
i ,k denotes the number of cycles accumulation-service for queue 2 in [ t k ( e, i) Ú T,
n
n,2
n,2
z k ( e, i) Ú T], s 0 (i, k), s 1 (i, k), ··· denote the switchover times from queue 2 to
n,2
n,2
n,2
queue 1 that start at n k n,2
0 (i, k), n k 1 (i, k ), . . . . We note again that s 0 (i, k ), s 1 (i, k),
n,2
. . . are independent and distributed as s 1 .
n ,2
Let v n ,1 (which is q n,1
(which is q n,2
T in the notation of Lemma 5.2) and v
T in Lemma
5.2) denote the number of respective switchovers from queue 1 to queue 2 and from queue
2 to queue 1 started in [0, nT]. By the definitions above (recall in particular that a nt Å 0
if the server is switching over at t), for k ¢ 1, 0 ° i ° N,
q n,1
i,k
q n,2
i,k
n,2
∑ s n,1
j (i, k) / ∑ s j (i, k) °
jÅ0
jÅ0
*
n ( z kn ( e,i ) ÚT )
n ( t kn ( e,i ) ÚT )
(1 0 a tn )dt ° ( max s jn,1 / max s jn,2 )
1° j°v n,1
q n,1
i,k/1
n
k
·1( t ( e, i) õ T ) /
1° j°v n,2
n,2/1
q i,k
∑ s (i, k) / ∑ s jn,2 (i, k),
n,1
j
jÅ0
jÅ0
so, since V nt √ [ai ( e ) 0 e, ai ( e ) / e] on [ t nk ( e, i), z nk ( e, i)) and b nt Å 1 0 a nnt , we get
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
1
q_
n
S
n,1
q i,k
n,2
q i,k
jÅ0
jÅ0
n,2
(ai ( e ) 0 e ) ∑ s n,1
j (i, k) / ∑ s j (i, k)
q_
° n
*
z nk ( e,i ) ÚT
t nk ( e,i ) ÚT
(5.40)
D
b nt V nt dt
1
° (ai ( e ) / e )·1( t nk ( e, i) õ T ) q_ ( max s n,1
/ max s n,2
j
j )
n,1
1°
j°v
1° j°v n,2
n
1
/ q_ (ai ( e ) / e )
n
S∑
q n,1
i,k/1
n,2/1
q i,k
s (i, k) / ∑ s n,2
j (i, k)
n,1
j
jÅ0
jÅ0
D
.
Introduce
S
P
P
n,1
n,2
q i,k
q i,k
1
A˜ nk ( e, i) Å q_ (ai ( e ) 0 e ) ∑ s jn,1 (i, k) / ∑ s jn,2 (i, k)
jÅ0
jÅ0
n
(5.41)
1
A˜ nk ( e, i) Å q_ (ai ( e ) / e )
n
(5.42)
S
˜ n,1
q
i,k/1
n,2/1
˜ i,k
q
jÅ0
jÅ0
D
,
∑ s jn,1 (i, k) / ∑ s jn,2 (i, k)
D
/ (ai ( e ) / e )·1( t nk ( e, i) õ T )ṽ n ,
P n,1
˜ n,2
P n,2
where q˜ n,1
i ,k and q i ,k are defined by (4.8) and (4.9) respectively, q i ,k and q i ,k denote their
counterparts for queue 2 and
1
ṽ n Å q_ ( max s jn,1 / max s jn,2 ).
1° j°v n,2
n 1° j°v n,1
It was shown in the proof of Lemma 5.2 (see (5.7)) that P[( v n , l /n) ú (T / 1)/dl ] r 0,
l Å 1, 2, as n r ` . Then, using (2.6) and Lemma 3.1, we get
P
ṽ n r 0 (n r ` ).
(5.43)
Also, inequalities (5.40) and (4.10) (an analogue of the latter holds obviously for queue
2 as well) yield
q_
A˜ nk ( e, i) ° n
(5.44)
*
z kn ( e,i ) ÚT
t nk ( e,i ) ÚT
b nt V nt dt ° AP nk ( e, i).
Next, (4.18), (4.16), and (4.17), and their analogues for queue 2 imply, in view of (4.4)
and Lemma 2.2 in Coffman et al. (1995), that
(5.45)
S
1
q_
n
D
d
r
q˜ n,l
i ,k
0°i°N,
k¢1,lÅ1,2
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zk ( e, i) Ú T 0 tk ( e, i) Ú T
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(5.46)
S
1
q_
n
D
d
r
qP n,l
i ,k
0°i°N,
k¢1,lÅ1,2
S
zk ( e, i) Ú T 0 tk ( e, i) Ú T
ai ( e ) / e
S
1
1
/
rl 1 0 rl
DD
01
.
0°i°N,
k¢1,lÅ1,2
Hence, by (5.41) – (5.43), Lemma 5.1, the equality r1 / r2 Å 1 and again Lemma 2.2
in Coffman et al. (1995),
d
(A˜ nk ( e, i) ) 0°i°N, r (A˜ k ( e, i)) 0°i°N, ,
(5.47)
k¢1
k¢1
and
d
(AP nk ( e, i) ) 0°i°N, r (AP k ( e, i)) 0°i°N, ,
(5.48)
k¢1
k¢1
where
ai ( e ) 0 e
( zk ( e, i) Ú T 0 tk ( e, i) Ú T ) r1r2 (d1 / d2 ),
ai ( e ) / e
(5.49)
Ãk ( e, i) Å
(5.50)
ai ( e ) / e
AP k ( e, i) Å
( zk ( e, i) Ú T 0 tk ( e, i) Ú T ) r1r2 (d1 / d2 ).
ai ( e ) 0 e
Introduce
`
(5.51)
N
AP n ( e ) Å ∑ ∑ AP kn ( e, i),
kÅ1 iÅ1
kÅ1 iÅ0
`
(5.52)
`
N01
A˜ n ( e ) Å ∑ ∑ A˜ kn ( e, i),
`
N01
˜ e ) Å ∑ ∑ A˜ k ( e, i),
A(
N
P e ) Å ∑ ∑ AP k ( e, i).
A(
kÅ1 iÅ1
kÅ1 iÅ0
Note that the sums are P-a.s. finite (use (4.3) for the second line), so that the variables
above are well defined.
Relations (5.47) and (5.48) imply, by the continuous mapping theorem, that for every
M Å 1, 2, . . . ,
M N01
M N01
kÅ1 iÅ1
kÅ1 iÅ1
d
∑ ∑ AP kn ( e, i) r ∑ ∑ AP k ( e, i),
M
N
d
M
N
∑ ∑ A˜ nk ( e, i) r ∑ ∑ A˜ k ( e, i)
kÅ1 iÅ0
kÅ1 iÅ0
as n r ` .
On the other hand, in a manner similar to (4.25),
SZ
lim lim P
Mr` nr`
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M N01
Z D
∑ ∑ AP nk ( e, i) 0 AP n ( e ) ú 0 Å 0,
kÅ1 iÅ1
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
SZ
Mr`
Z D
M N01
P e ) ú 0 Å 0,
∑ ∑ AP k ( e, i) 0 A(
lim P
kÅ1 iÅ1
and Theorem 4.2 in Billingsley (1968) yields
d
(n r ` ).
d
(n r ` ).
P e)
AP n ( e ) r A(
(5.53)
Similarly,
˜ e)
A˜ n ( e ) r A(
(5.54)
Next, in analogy with the proof of (4.29), we get in view of (4.2) that, as e r 0,
0
˜
° K)dt.
* 1( d ° V(t)
T
P
˜ e ) r r1r2 (d1 / d2 )
A(
(5.56)
˜
° K)dt,
* 1( d ° V(t)
T
P
P e ) r r1r2 (d1 / d2 )
A(
(5.55)
0
Also, it is not difficult to see, using the definitions of t nk ( e, i) and z nk ( e, i), that
_
∑ ∑ n
`
N01 q
kÅ1 iÅ1
*
z nk ( e,i ) ÚT
t nk ( e,i ) ÚT
q_
b nt V nt dt ° n
* b V ·1( d ° V
T
n
t
n
t
_
*
n
t
° K)dt
0
`
°
N q
∑ ∑ n
kÅ1 iÅ0
z kn ( e,i ) ÚT
t nk ( e,i ) ÚT
b nt V nt dt,
so by (5.44) and (5.51)
q_
A˜ n ( e ) ° n
* b V ·1( d ° V
T
n
t
n
t
n
t
0
° K)dt ° AP n ( e ).
The latter and (5.53) – (5.56) imply (5.37) by Lemma 2.3 in Coffman, Puhalskii, and
h
Reiman (1995).
6. Proofs of theorems.
PROOF
(6.1)
OF
THEOREM 2.1.
By (5.23) and (5.21),
q_
(VV nt ) 2 / d nt Å (VV 0n ) 2 / 2 n ( r n 0 1)
* VV ds / (2d / s )t / 2 * VV
t
t
n
s
2
0
0
where
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n
s0
dM ns ,
POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
q_
d nt Å 2 n
*
t
q_
e ns ·1 (V ns Å 0)ds / 2 n
0
* e b ·1(V
297
t
n
s
n
s
n
s
ú 0)ds
0
q_
/ 2dt 0 2 n
* b V ds / s t 0 ∑ ( DM ) .
t
n
s
n
s
2
n 2
s
0
s°t
By (5.16), (5.17), Lemma 5.5 and Lemma 5.8,
P
supÉd ns É r 0(n r ` ),
(6.2)
t ú 0.
s°t
We denote the left-hand side of (6.1) by X nt . Let X n Å (X nt , t ¢ 0). We next prove that
X n converges in distribution to X as n r ` . By (6.1), the process X n is an F n-locally
square-integrable semimartingale (Jacod and Shiryaev (1987), Ch. II, §2b)). The process
X as defined by (2.9) is a locally square-integrable semimartingale as well with respect
to the filtration generated by it (Jacod and Shiryaev (1987), III.2.12). We prove the
convergence by applying Theorem IX.3.48 in Jacod and Shiryaev (1987) which gives
conditions for convergence in distribution of a sequence of semimartingales to a semimartingale in terms of their predictable characteristics (Jacod and Shiryaev (1987), Ch.
II, §2). Therefore our first step is to identify the characteristics. Let B * n Å (B *t n , t ¢ 0)
˜ *t n , t ¢ 0) denote its
denote the first characteristic without truncation of X n , let C̃ * n Å (C
n
modified second characteristic without truncation and let n Å ( n n (ds, dx)) denote its
predictable measure of jumps (Jacod and Shiryaev (1987), II.2.29, IX.3.25). Then by
(6.1)
q_
B *t n Å 2 n ( r n 0 1)
(6.3)
* VV ds / (2d / s )t,
t
n
s
2
0
* (VV ) d » M … .
t
˜ *t n Å 4
C
(6.4)
n 2
s
n
s
0
Define next, for a Å ( at , t ¢ 0), an element of the Skorohod space D[0, ` ),
* ( a Û 0)
t
(6.5)
Bt ( a ) Å 2c
s
1/2
ds / (2d / s 2 )t,
t ¢ 0,
0
* ( a Û 0)ds,
t
Ct ( a ) Å 4s 2
(6.6)
s
t ¢ 0,
0
(6.7)
n([0, t], G )( a ) Å 0,
t ¢ 0,
G is a Borel subset of R,
and let B( a ) Å (Bt ( a ), t ¢ 0), C( a ) Å ( Ct ( a ), t ¢ 0) and n( a ) Å ( n( dt, dx )( a )).
According to the definition of X in (2.9), its triplet of predictable characteristics is (B( X),
C( X), n(X)). Since X is continuous, this triplet does not depend on a truncation function;
in particular the triplet without truncation is the same.
Stated in another way, the distribution of X is the unique solution to the martingale
problem associated with ( H , X) and ( L ( X0 ); B, C, n ), in the sense of definition III.2.4
of Jacod and Shiryaev (1987), where H denotes the s-field generated by X0 and L (X0 )
denotes the distribution of X0 .
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
Define next, as in IX.3.38 of Jacod and Shiryaev (1987), for a ¢ 0,
(6.8)
Sa ( a ) Å inf(t : ÉatÉ ¢ a or Éat0É ¢ a),
(6.9)
n
S an Å inf(t : É X tn É ¢ a or É X t0
É ¢ a),
where at0 denotes the left-hand limit of a at t. Let also (Var B)t ( a ) denote the total
variation of B( a ) on [0, t] and C1 (R) denote the set of continuous bounded functions
g : R r R that are equal to 0 in a neighborhood of 0.
By Theorem IX.3.48 of Jacod and Shiryaev (1987), in order to prove that the X n
converge in distribution to X, we may check the following conditions (note that since X
has no jumps, in the notation of the theorem, B * Å B and C̃ * Å C):
(i) The local strong majorization hypothesis: for all a ¢ 0, there is an increasing
continuous and deterministic function F( a) Å (Ft ( a), t ¢ 0) such that the stopped protÚS ( a )
cesses ((Var B) tÚSa ( a ) ( a ), t ¢ 0), (CtÚSa ( a ) ( a ), t ¢ 0) and ( *0 a *R É xÉ2n( ds, dx )( a ),
t ¢ 0) are strongly majorized by F( a) for all a √ D[0, ` ).
(ii) The local condition on big jumps: for all a ¢ 0, t ú 0,
*
tÚSa ( a )
lim
sup
br` a√ D[0,` )
* É xÉ 1(É xÉ ú b) n(ds, dx)( a ) Å 0.
2
0
R
(iii) Local uniqueness for the martingale problem associated with ( H , X) and ( L (X0 );
B, C, n ) (see Jacod and Shiryaev (1987), III.2).
t
(iv) The continuity condition: the maps a r Bt ( a ), a r Ct ( a ) and a r *0 *R g(x) n( ds,
dx) are continuous
for the Skorohod topology on D[0, ` ) for all t ú 0 and g √ C1 ( R).
d
(v) X n0 r X0 .
P
tÚS n
tÚS n
(vi) [ dloc 0 R/ ] *0 a *R g(x ) n n (ds, dx) 0 *0 a *R g(x) n(ds, dx )(X n ) r 0 for all
t ú 0, a ú 0 and g √ C1 (R);
P
n
n
n 0 BsÚS n (X )É r 0 for all t ú 0, a ú 0;
[sup 0 b*loc ] supÉB *sÚS
a
a
s°t
P
n
n
˜ *tÚS
n 0 CtÚS n (X ) r 0 for all t ú 0, a ú 0;
[ g*loc 0 R/ ] C
a
a
tÚS na
2
(6.9a) lim lim P( *0 *R É xÉ 1(É xÉ ú b) n n ( ds, dx) ú e ) Å 0 for all t ú 0, a ú 0 and
br` nr`
e ú 0.
This last condition is Equation (3.49) in Jacod and Shiryaev (1987).
We now check these 9 conditions in order. We have, by (6.5) – (6.8), for s õ t,
(Var B) tÚSa ( a ) ( a ) 0 (Var B) sÚSa ( a ) ( a ) ° 2É cÉa 1 / 2 (t 0 s) / (2d / s 2 )(t 0 s),
CtÚSa ( a ) ( a ) 0 CsÚSa ( a ) ( a ) ° 4s 2a(t 0 s),
*
tÚSa ( a )
0
* É xÉ n(du, dx)( a ) 0 *
sÚSa ( a )
2
R
0
* É xÉ n(du, dx)( a ) Å 0.
2
R
This verifies condition (i) with Ft (a) Å K(a) t, where K( a) is large enough.
Condition (ii) holds since n Å 0. Condition (iii) (local uniqueness) holds by Theorem
III.2.40 of Jacod and Shiryaev (1987) since the equation
* (Y Û 0)
t
Yt Å 2c
s
ds / (2d / s 2 )t / 2s
0
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* (Y Û 0)
t
1/2
Wednesday May 06 03:12 PM
s
1/2
dWs / x,
0
INF–MOR
0014
POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
299
where W Å (Wt , t ¢ 0) is a Wiener process, has a unique (weak) solution (Ikeda and
Watanabe (1989), Chapter IV, Theorems 1.1, 2.4, and 3.2, Example 8.2) for any x √ R,
and since one can set, in the conditions of Theorem III.2.40 of Jacod and Shiryaev (1987),
pt B Å B, ptC Å C, ptn Å n Å 0.
Condition (iv) follows from (6.5) – (6.7) by the argument of the proof of Theorem
6.2.2 in Liptser and Shiryaev (1989) since Skorohod convergence implies convergence
at continuity points of the limit.
d
Condition (v) holds by the assumption V n0 r V0 , (5.21), Lemma 5.5 and (6.2).
Consider condition [ dloc 0 R/ ] under (vi). Since n Å 0, it is enough to prove that
* * É g(x)Én (ds, dx) r 0.
t
P
n
0
R
Since, by the definition of C1 (R), for some e ú 0, g(x ) Å 0 if É xÉ õ e, and g(x ) is
bounded,
the latter integral converges in probability to zero as n r ` if n n ([0, t], {É xÉ
P
ú e}) r 0. By Lemma 5.5.1 in Liptser and Shiryaev (1989), this is implied by
P
supÉD X sn É r 0(n r ` ),
(6.10)
t ú 0.
s°t
By the definition of X n , ÉD X ns É ° DÉVV ns É2 / ÉDd ns É, and (6.10) holds by (6.2) and the
C-tightness of VV n (use Proposition VI.3.26 in Jacod and Shiryaev (1987)).
Next, we check condition [sup 0 b*loc ] under (vi). By the definition of X n , (6.3) and
(6.5),
q_
supÉ B sÚS
* n na 0 BsÚS na (X n )É ° 2É n ( r n 0 1) 0 cÉ
s°t
* ÉVV É ds
t
n
s
0
* ÉVV
t
/ 2É cÉ
0
n
s
0 (((VV sn ) 2 / d sn ) Û 0) 1 / 2É ds
q_
° 2É n ( r n 0 1) 0 cÉt supÉVV sn É / 2É cÉt supÉd sn É1 / 2 ,
s°t
s°t
q_
and the latter converges inP probability to 0 as n r ` , since n ( r n 0 1) r c , {VV n , n
¢ 1} is tight and supÉd ns É r 0 by (6.2).
s°t
Now consider condition [ g*loc 0 R/ ] under (vi). By (6.4), (6.6) and the definition of
Xn
n
n
˜ *tÚS
n 0 CtÚS n (X )É
ÉC
a
a
(6.11)
Z*
s
° 4 sup
s°t
0
* (VV ) duZ / 4s t supÉd É.
s
(VV un ) 2 d » M n …u 0 s 2
n 2
u
2
n
s
s°t
0
The last term converges in probability to 0 by (6.2). Since {(VV n ) 2 , n ¢ 1} is C-tight,
we have, for h ú 0,
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
lim lim P(
dr0 nr`
sup
É(VV nu ) 2 0 (VV n£ ) 2É ú h ) Å 0,
Éu0£Éõd,u,£°t
which, in view of the second assertion of Lemma 5.5, is seen to yield (the idea of the
proof is as in (Billingsley (1968), Problem 8, §2))
Z*
* (VV ) duZ r 0.
s
sup
s°t
0
s
(VV un ) 2 d » M n …u 0 s 2
n 2
u
P
0
In view of (6.11), this concludes the verification of [ g*loc 0 R/ ].
Finally, consider condition (6.9a) under (vi). Define
* * É xÉ ·1(É xÉ ú b) n (ds, dx),
t
L̃ nt Å
2
0
n
t ¢ 0.
R
We have for e ú 0, A ú 0,
(6.12)
S*
P(L˜ nt ¢ e ) ° P(supÉVV ns É ú A) / P
s°t
D
t
0
1(VV ns0 ° A)dL˜ ns ú e .
The first term on the right goes in probability to 0 as n r ` and A r ` by the tightness
of VV n .
Next, letting
g n,1
Å 1n t n,1
t
 n ( l1t/1 )  ,
g n,2
Å 1n t n,2
t
 n ( l2t/1 )  ,
n,l
we have for the second term on the right of (6.12), since A n,l
nt ¢  n( llt / 1)  when g t
õ t, l Å 1, 2,
S*
t
n
1(VV s0
° A)dL˜ sn ú e
P
0
S
S*
°P
D
1 n,1
1
A nt ú l1t / 1 0
n
n
n,2
tÚg n,1
t Úg t
/P
0
D S
/P
1 n,2
1
A nt ú l2t / 1 0
n
n
D
D
1(VV ns0 ° A)dL˜ sn ú e .
Again the first two terms on the right tend to 0 as n r ` by Lemma 5.1. It is thus left to
prove that the last term on the right tends in probability to 0 as n r ` .
Since n n ( ds, dx) is the predictable measure of jumps of X n , by (6.1) the process L̃ n
Å (L˜ nt , t ¢ 0) is the F n-compensator of the process L n Å (L nt , t ¢ 0) defined by
(6.13)
n
n
L tn Å 4 ∑ (VV s0
) 2 ( DM sn ) 2·1(2ÉVV s0
\DM sn É ú b).
0õs°t
t
n
Accordingly, the process ( *0 1(VV s0
° A)dL˜ sn , t ¢ 0) is the F n-compensator of the process
t
n
n
V
( *0 1(V s0 ° A)dL s , t ¢ 0). Therefore, by (6.13) and, since, by Lemma 5.4, g tn,1 and
g n,2
are F n-stopping times, the Lenglart-Rebolledo inequality implies that, for h ú 0,
t
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POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
S*
tÚg tn,1Úg tn,2
P
0
S
S
D
S
1
b
h/E
sup
4A 2ÉDM sn É2·1 ÉDM ns É ú
n,1
n,2
e
2A
s°tÚg t Úg t
°
(6.14)
1(VV ns0 ° A)dL˜ ns ú e
∑
/ P 4A 2
S
( DM sn ) 2·1 ÉDM sn É ú
0õs°tÚg tn,1Úg tn,2
DD
D D
b
¢h .
2A
n,2
By the definitions of M n (see (5.19)) and g n,1
t , gt ,
E
S
ÉDM ns É21 ÉDM sn É ú
sup
s°tÚg tn,1Úg tn,2
b
2A
D
S
6
b q_
n n,1 2
n,1
n n,1
° E
sup ( h n,1
0
r
j
)
·
1
Éj
0
r
h
É
ú
n
i
1 i
i
1 i
n i°  n ( l1t/1 ) 
4A
D
S
6
b q_
n n,2 2
n,2
n n,2
/ E
sup ( h n,2
0
r
j
)
·
1
Éj
0
r
h
É
ú
n
i
2 i
i
2 i
n i°  n ( l2t/1 ) 
4A
S
2
n,1
n n,1
° 6( l1t / 1)E( h n,1
0 r n1 j n,1
i
i ) ·1 Éj 1 0 r 1 h 1 É ú
b q_
n
4A
S
D
b q_
n
4A
2
n,2
n n,2
/ 6( l2t / 1)E( h n,2
0 r n2 j n,2
i
i ) ·1 Éj 1 0 r 2 h 1 É ú
D
D
,
which tends to 0 as n r ` by (2.4) and (2.5). The third term on the right of (6.14) is
not greater than
S F
P 4A 2
/
6
n
6
n
S
 n ( l1t/1 ) 
∑
( j in,1 0 r 1n h in,1 ) 2·1 Éh in,1 0 r 1n j in,1 É ú
iÅ1
S
 n ( l2t/1 ) 
∑
2
n,2
( h n,2
0 r n2 j n,2
0 r n2 h in,2w ú
i
i ) ·1 Éj i
iÅ1
b q_
n
4A
b q_
n
4A
D
DG D
¢h ,
which again tends to 0 by (2.4) and (2.5). Therefore, by (6.14),
S*
n,2
tÚg n,1
t Úg t
lim P
nr`
0
D
h
1(VV ns0 ° A)dL˜ ns ú e ° ,
e
which completes the check of (6.9a) since h is arbitrary.
Thus, all the conditions of Theorem IX.3.48 in Jacod and Shiryaev (1987) hold and,
by the theorem, {X n , n ¢ 1} converges in distribution to X, which by Lemma 2.1 is
distributed as V 2 . Since (5.21), Lemma 5.5, and (6.2) imply
P(supÉ(V ns ) 2 0 X sn É ú d ) r 0(n r ` ),
t ú 0,
d ú 0,
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E. G. COFFMAN, JR., A. A. PUHALSKII AND M. I. REIMAN
d
d
we have that (V n ) 2 r V 2 , and hence that V n r V since all the processes are nonnegative.
h
PROOF OF THEOREM 2.2.
h
Lemma 5.7.
The theorem follows by Theorem 2.1, Theorem 4.1 and
PROOF OF THEOREM 2.3.
has the form
By (2.12) and the definition of V̆ n , a basic equation for V̆ n
q_
V˘ nt Å V 0n / B˘ tn / nrV n
*
t
q_
1(V˘ sn Å 0)ds / n
0
*
t
0
q_
1(V˘ sn ú 0)(rV n 0 r n ( nV˘ sn ))ds,
where
S nnt 0 rV nnt
.
B˘ nt Å
q_
n
The equation is similar to Equation (5.2) and the proof goes exactly the way it does for
{V n , n ¢ 1}. We first prove the C-tightness of {V̆ n , n ¢ 1} by following the same steps
as in §5. The only difference is that the K n ,e are defined this time by
*
q_
K n,e
Å
n
t
t
0
q_
1(V˘ sn ú e )(rV n 0 r n ( nV˘ sn ))ds,
and Lemma 5.2 is trivial because of (r1).
As can be seen from the proof of Theorem 2.1 in §6, after the C-tightness has been
proved, the only additional fact that is needed is the convergence
q_
n
* V b ds r dt.
t
n
s
n
s
P
0
In the case of V̆ n , this amounts to proving that
q_
n
*
t
0
q_
P
V˘ ns (rV n 0 r n ( nV˘ ns ))ds r dt.
This turns out to be a simple consequence of (r1), (r2) and an analogue of Lemma 5.7
for V̆ n which is proved in the same way. According to the latter result, for h ú 0,
S*
t
(6.15)
lim lim P
er0 nr`
D
1(V˘ ns õ e )ds ú h Å 0.
0
Then, for e ú 0,
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POLLING SYSTEMS IN HEAVY TRAFFIC: A BESSEL PROCESS LIMIT
SZ *
q_
P
Z D
q_
V˘ ns (rV n 0 r n ( nV˘ ns ))ds 0 dt ú h
t
n
0
S*
t
°P
0
q_
q_
h
É nV˘ ns (rV n 0 r n ( nV˘ ns )) 0 dÉ·1(V˘ ns õ e )ds ú
2
S*
t
/P
0
S
303
D
q_
q_
h
É nV˘ ns (rV n 0 r n ( nV˘ sn )) 0 dÉ·1(V˘ sn ¢ e )ds ú
2
* 1(V˘
t
° P (sup x(rV n 0 r n (x)) / d)
x,n
n
s
õ e )ds ú
0
h
2
D
D
/ 1(t sup_ É x(rV n 0 r n (x)) 0 dÉ ú h2 ).
q
x¢ n e
The probability on the right-hand side tends to 0 as n r ` and e r 0 by (r2) and (6.15),
h
and the indicator goes to 0 as n r ` by (r1).
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,
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ups. TR 98-1. Industrial and Operations Engineering Dept., University of Michigan, Ann Arbor, MI.
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