DTIC ADA439467: On a Random Sum Formula for the Busy Period of the M/G/Infinity Queue With Applications

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ECHNICAL ESEARCH EPORT
On a Random Sum Formula for the Busy Period of the M|G|
Infinity Queue with Applications
by Armand M. Makowski
CSHCN TR 2001-4
(ISR TR 2001-9)
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On a Random Sum Formula for the Busy Period of the M/G/ Infinity
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Queue with Applications
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unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
On a random sum formula
for the busy period
M G
of the j j1 queue
with applications
y
Armand M. Makowski
[email protected]
[phone]
Abstract
A random sum formula is derived for the forward recurrence time
associated with the busy period length of the MjGj1 queue. This
result is then used to (i) provide a necessary and su(cid:14)cient condition
for the subexponentiality of this forward recurrence time, and (ii)
establishastochasticcomparisonintheconvex increasing(variability)
orderingbetween the busyperiodsintwo MjGj1 queues withservice
times comparable in the convex increasing ordering.
Key words: Random sums, MjGj1 queues, busy period, subexponen-
tiality, stochastic orderings
AMS: 60K25, 60E15
1 Introduction
Random sums, and geometric random sums in particular, are a common
occurence in applied probability models [4, 6]. For instance, it is well known
y
Electrical Engineering Department and Institute for Systems Research, University of
Maryland, College Park, MD 20742. The work of this author was supported partially
through NSF Grant NSFD CDR-88-03012, NASA Grant NAGW77S and the Army Re-
search Laboratoryunder Cooperative Agreement No. DAAL01-96-2-0002.
1
that the stationary waiting time of a stable MjGIj1 queue with Poisson
arrival rate (cid:21) and generic service time (cid:27) can be represented (in distribution)
as a geometric sum of i.i.d. rvs distributed like the forward recurrence time
?
(cid:27) associated with (cid:27) [7]. A similar representation holds for the stationary
waiting time of a stable GIjGIj1 queue in terms of ladder height rvs [10].
Such random sum representations have proved useful for establishing various
properties of interest [4, 6].
Perhaps less well known is the fact that similar geometric sums can also
be found in the context of MjGIj1 queues. Indeed, consider a standard
MjGj1queuewitharrivalrate(cid:21)andgenericservicetime(cid:27),andletB denote
its generic busy period length. Using some classical results on the Laplace
?
transform of B, we show that the forward recurrence time B associated
with the busy period of an MjGIj1 can be represented as a geometric sum
of i.i.d. rvs whose common distribution is derived from that of the forward
?
recurrence time (cid:27) . This random sum representation is presented in Section
2, and its proof is given in Section 5.
We give two applications for this representation result: In Section 3, we
? ?
show that the subexponentiality of (cid:27) is equivalent to that of B , with a
simple relation between the tail of their distributions. This result was orig-
inally derived by Boxma [1] for regularly varying (cid:27), and further generalized
in the form given here (but with a di(cid:11)erent proof) by Jelenkovic and Lazar
?
[5] to the case where (cid:27) is subexponential. In Section 4, we investigate the
monotonicitypropertiesofMjGIj1queues underthe(increasing)convex or-
dering. Using the random sum representation we show that the busy period
distribution is monotone in the increasing convex ordering when the service
time distribution is increased in the increasing convex ordering. To the best
of the author’s knowledge this result is new [14, p. 147].
A word on the notation used in this paper: For any integrable IR+{valued
?
rv X, the forward recurrence time X is de(cid:12)ned as the rv with integrated
tail distribution given by
1
? 1
P[X > x] := P[X > t]dt; x (cid:21) 0: (1)
E[X] Zx
We shall (cid:12)nd it useful to use the equivalent representation
+
? E[(X (cid:0)x) ]
P[X > x] := ; x (cid:21) 0 (2)
E[X]
2
+
(where we write x = max(x;0) for any scalar x). For mappings f;g : IR+ !
f(x)
IR, the relation f(x) (cid:24) g(x) is understood as limx!1 g(x) = 1, the quali(cid:12)er
(x ! 1) being omitted for the sake of notational simplicity.
2 A random sum in the MjGj1 queue
Consider a standard MjGj1 queue with arrival rate (cid:21) and generic service
time (cid:27); we refer to this queueing system as the MjGj1 queue ((cid:21);(cid:27)). Let B
denote its generic busy period, i.e., the length of time that elapses between
the arrival of a customer (cid:12)nding an empty system and the departure of the
(cid:12)rst customer thereafter which leaves the system empty.
?
In Section 5 we show that the forward recurrence timeB associated with
B admits a random sum representation: In order to present this result, let
(cid:23) denote the IN{valued rv which is geometrically distributed according to
‘(cid:0)1
P[(cid:23) = ‘] = (1(cid:0)K)K ; ‘ = 1;2;::: (3)
with
(cid:0)(cid:26)
(cid:26) := (cid:21)E[(cid:27)] and K := 1(cid:0)e : (4)
Next, we introduce the IR+-valued rv U distributed according to
1 (cid:0)(cid:26)P[(cid:27)?(cid:20)x]
P[U (cid:20) x] := (cid:16)1(cid:0)e (cid:17); x (cid:21) 0 (5)
K
?
where (cid:27) is the forward recurrence time associated with the generic service
time (cid:27). We are now ready to formulate the key observation of the paper:
Theorem 1 Consider a sequence of IR+{valued i.i.d. rvs fUn; n = 1;2;:::g
distributed according to (5), and an IN{valued rv (cid:23) distributed according to
(12). Then, with the sequence fUn; n = 1;2;:::g taken to be independent
of the rv (cid:23), it holds that
?
B =st U1 +:::+U(cid:23): (6)
where =st denotes equality in distribution.
In analogy with standard results for the GIjGIj1 queue [10], it is natural
to wonder whether the forward recurrence time associated with the busy
3
period in the GIjGIj1 queue also admits a representation as a geometric
randomsumofi.i.d. rvswhosedistributionisnowderivedfromthatofladder
height rvs. To the best of the author’s knowledge no results along these lines
are known.
In the process of establishing Theorem 1 in Section 5 we shall show that
K
E[B] = : (7)
(cid:21)(1(cid:0)K)
We also note from (4) and (5) that
(cid:0)
(cid:26)
e (cid:26)P[(cid:27)?>x]
P[U > x] = e (cid:0)1 ; x (cid:21) 0: (8)
K (cid:16) (cid:17)
Moreover, using (2) we see that (8) can be rewritten as
(cid:0)
(cid:26)
e (cid:21)E[((cid:27)(cid:0)x)+]
P[U > x] = e (cid:0)1 ; x (cid:21) 0: (9)
K (cid:18) (cid:19)
M G
3 Subexponentiality in the j j1 queue
Webeginwithsomestandardde(cid:12)nitionsandfactsconcerningsubexponential
rvs [2, 3]: The IR+-valued rv X is said to have a subexponential tail, denoted
X 2 S, if
P[X1 +:::+Xn > x] (cid:24) nP[X > x]; n = 1;2;::: (10)
where fXn; n = 1;2;:::g denotes a sequence of i.i.d. rvs, each distributed
like X. In fact, (10) holds for all n = 1;2;::: if and only if it holds for some
n (cid:21) 2. Under appropriate conditions, the equivalences (10) can be extended
to random sums [3, Thm. 1.3.9, p. 45] (and [3, Thm. A3.20, p. 580]).
Proposition 1 Let the IN{valued rv N be independent of the sequence of
i.i.d. rvs fX;Xn; n = 1;2;:::g. If X 2 S, then X1 +:::+XN 2 S with
P[X1 +:::+XN > x] (cid:24) E[N]P[X > x] (11)
N
provided E z < 1 for some z > 1.
h i
4
Of particular interest for the discusssion here is the case when N is dis-
tributed according to the geometric distribution
(cid:0)
‘ 1
P[N = ‘] = (1(cid:0)p)p ; ‘ = 1;2;::: (12)
(cid:0)
1
for some 0 < p < 1. Standard calculations yield E[N] = (1(cid:0)p) , and
1
N ‘(cid:0)1 ‘ (1(cid:0)p)z (cid:0)1
Ehz i = X(1(cid:0)p)p z = ; jzj < p (13)
‘=1 1(cid:0)pz
(cid:0)1
with p > 1. Thus, Proposition 1 applies, in fact, can be strenghtened as
follows [3, Cor. A3.21, p. 581]:
Proposition 2 Let the IN{valued rv N be independent of the sequence of
i.i.d. rvs fX;Xn; n = 1;2;:::g, and assume N to be distributed according
to (12). Then, X 2 S if and only if X1 +:::+XN 2 S, in which case
(cid:0)1
P[X1 +:::+XN > x] (cid:24) (1(cid:0)p) P[X > x] (14)
The main result of this section is the following
Proposition 3 Consider a standard MjGj1 queue ((cid:21);(cid:27)) with generic busy
? ?
period rv B. We have B 2 S if and only if (cid:27) 2 S, in which case
? (cid:26) ?
P[B > x] (cid:24) (cid:0)(cid:26)P[(cid:27) > x]: (15)
1(cid:0)e
Proof. Combining Theorem 1 with Proposition 2, we already get that
?
B 2 S if and only U 2 S, in which case
? (cid:0)1
P[B > x] (cid:24) (1(cid:0)K) P[U > x]: (16)
?
Next, with (8) in mind, we observe that limx!1P[(cid:27) > x] = 0, so that
(cid:26)P[(cid:27)?>x] ? ?
e = 1+(cid:26)P[(cid:27) > x]+o(P[(cid:27) > x]):
Hence, from (8) we get
(cid:0)(cid:26)
e ? ?
P[U > x] = ((cid:26)P[(cid:27) > x]+o(P[(cid:27) > x]));
K
5
and the conclusion (cid:0)
(cid:26)
(cid:26)e ?
P[U > x] (cid:24) P[(cid:27) > x] (17)
K
follows. Therefore, since S is closed under tail-equivalence [3, Lemma A3.15,
?
p. 572], we get U 2 S if and only if (cid:27) 2 S, and we complete the proof of
(15) by injecting this last fact with (17) into the equivalence (16).
M G
4 Orderings in the j j1 queue
For IR{valued random variables X and Y, we say that X is smaller than Y
in the strong stochastic (resp. convex, increasing convex) ordering if
E[’(X)] (cid:20) E[’(Y)] (18)
for all mappings ’ : IR ! IR which are monotone increasing (resp. convex,
increasing and convex) provided the expectations in (18) exist. In that case
we write X (cid:20)st Y (resp. X (cid:20)cx Y, X (cid:20)icx Y). Additional material on these
orderings can be found in the monographs [11, 13, 14]. The following result
is well known [14, Prop. 2.2.5, p. 45].
Proposition 4 Let the IN{valued rv N be independent of the sequences of
i.i.d. rvs fX;Xn; n = 1;2;:::g and fY;Yn; n = 1;2;:::g. If X (cid:20)st Y, then
it holds that
X1 +:::+XN (cid:20)st Y1 +:::+YN (19)
Results similar to (19) hold mutatis mutandis under the weaker assump-
tions X (cid:20)cx Y and X (cid:20)icx Y. The main result of this section is the following
stochastic comparison.
Proposition 5 ConsidertwoMjGj1queues ((cid:21);(cid:27)1)and((cid:21);(cid:27)2)withgeneric
busy period rvs B1 and B2, respectively. If (cid:27)1 (cid:20)cx (cid:27)2, then B1 (cid:20)cx B2.
Throughout, for each i = 1;2, quantities associated with the MjGj1
queues ((cid:21);(cid:27)i) are subscripted by i.
Proof. Under the condition (cid:27)1 (cid:20)cx (cid:27)2, E[(cid:27)1] = E[(cid:27)2], whence (cid:26)1 = (cid:26)2
6
+ +
and K1 = K2, and the inequalities E[((cid:27)1 (cid:0)x) ] (cid:20) E[((cid:27)2 (cid:0)x) ] hold for
all x (cid:21) 0. Invoking (9), we immediately conclude that U1 (cid:20)st U2. Next,
applying Proposition 4 to the random sum representation (6), we see that
? ?
B1 (cid:20)st B2, namely
1 1
1 1
P[B1 > t]dt (cid:20) P[B2 > t]dt; x (cid:21) 0: (20)
E[B1] Zx E[B2] Zx
This is equivalent to
+ +
E[(B1 (cid:0)x) ] E[(B2 (cid:0)x) ]
(cid:20) ; x (cid:21) 0: (21)
E[B1] E[B2]
Finally,observefrom(7)andfromtheobservationsmadeabovethatE[B1] =
+ +
E[B2], so that E[(B1 (cid:0)x) ] (cid:20) E[(B2 (cid:0)x) ] for all x (cid:21) 0, and the desired
conclusion readily follows [14, Thm. 1.3.1, p. 9].
Theliteraturecontainsfewstochasticcomparisonresultsforin(cid:12)niteserver
queues; they dealmostlywiththenumberofcustomers, e.g., [14, Prop. 7.1.1,
p. 127], [14, Table 7.2, p. 147]. However, a simple coupling argument readily
leads to the following comparison:
Proposition 6 Consider twoMjGj1queues ((cid:21);(cid:27)1)and((cid:21);(cid:27)2)withgeneric
busy period rv B1 and B2, respectively. If (cid:27)1 (cid:20)st (cid:27)2, then B1 (cid:20)st B2.
Finally, we combine Propositions 5 and 6 with the characterization of the
convex increasing ordering provided in [9]:
Proposition 7 ConsidertwoMjGj1queues ((cid:21);(cid:27)1)and((cid:21);(cid:27)2)withgeneric
busy period rv B1 and B2, respectively. If (cid:27)1 (cid:20)icx (cid:27)2, then B1 (cid:20)icx B2.
5 A Proof of Theorem 1
Consider the process of particle counting as described in Chapter 6 of the
monographby Tak(cid:19)acs [15, p. 205]. Type II counters areequivalent toin(cid:12)nite
server queues if particles are interpreted as customers. So-called \registered"
particles [15, p. 205] are those particles which arrive at an instant when
there is no other particle present; in the in(cid:12)nite server queue context, such
7
a registered customer is a customer that initiates a busy period. Let the rv
R denote the length of time that elpases between the arrival epochs of two
consecutive registered particles, or equivalently, in the in(cid:12)nite server queue,
the time duration between the start of two consecutive busy periods.
Theorem 1 in [15, p. 210] provides a closed form expression for the
Laplace{Stieltjes transform of the rv R when customers arriva according to
a Poisson process: For the MjGj1 queue ((cid:21);(cid:27)), it holds that
(cid:0)(cid:18)R 1 1
E e = 1(cid:0) (cid:1) ; (cid:18) (cid:21) 0 (22)
(cid:21)+(cid:18) T((cid:18))
h i
with
1 t
T((cid:18)) := exp (cid:0)(cid:18)t(cid:0)(cid:21) P[(cid:27) > x]dx dt: (23)
0 0
Z (cid:18) Z (cid:19)
Lemma 1 With the notation (4){(5), it holds that
1 (cid:0)(cid:18)U
T((cid:18)) = 1(cid:0)KE e ; (cid:18) > 0: (24)
(cid:18)
(cid:16) h i(cid:17)
Proof. Fix (cid:18) (cid:21) 0. From (5) we note that
e(cid:0)(cid:21)E[(cid:27)]P[(cid:27)?(cid:20)t] = 1(cid:0)KP[U (cid:20) t]; t (cid:21) 0: (25)
Making use of this fact in the de(cid:12)nition (23), we (cid:12)nd
1 (cid:0)(cid:18)t (cid:0)(cid:21)E[(cid:27)]P[(cid:27)?(cid:20)t]
T((cid:18)) = e e dt
0
Z 1
(cid:0)(cid:18)t
= e (1(cid:0)KP[U (cid:20) t])dt
0
Z
1
1 (cid:0)(cid:18)t
= (cid:0)K e P[U (cid:20) t]dt
(cid:18) 0
Z (cid:0)(cid:18)t 1 1 (cid:0)(cid:18)t
1 e e d
= (cid:0)K P[U (cid:20) t] (cid:0) P[U (cid:20) t]dt
(cid:18) " (cid:0)(cid:18) #0 Z0 (cid:0)(cid:18) dt !
and the desired conclusion (24) follows from the fact P[U (cid:20) 0] = 0.
8