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Large sample inference in retrial queues
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical and Computer Modelling30 (1999)
197-206
www.elsevier.nl/locate/mcm
Large Sample Inference in Retrial Queues A. RODRIGO AND M. VAZQUEZ Departamento de Anaiisis Economico Facultad de Ciencias Economicas Universidad Complutense de Madrid Campus de Somosaguas, 28223 Madrid, Spain ececo03Qeis.ucm.es Abstract-we
analyze retrial queueing systems from a statistical viewpoint. We consider a
general G/G/l retrial queue where the flow of repeated attempts can be non-Markovian. We observe the operation of the system over a random time interval (O,T] focusing on the case where T is a departure epoch. We obtain some asymptotic relationship among the random variables involved, This allows us to get asymptotically Gaussianconsistent even when the system in nonergodic. estimators for the unknown parameters in a parametric context. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-G/G/l
retrial queue, Maximum likelihood estimation,
General retrials.
1. INTRODUCTION In communication networks customers do not usually wait in a queue when the service area is busy. They leave and try again some time later. We say that they join a new flow of repeated arrivals (calls) which is different from the primary flow. More precisely, if a customer finds the service area busy he “goes to an orbit” wherefrom he retries his call until he finds the service free. If the service area is free the customer starts to be served immediately, leaving the system after he has finished his service. The flow of repeated calls can be constant (see [1,2]) or depend on the number of customers in orbit (see the surveys by Falin [3,4]). Many articles have been published dealing with the theory of retrial systems but very few analyze these systems from a statistical viewpoint. Falin [5] and Rodrigo et al. 16) construct integral estimators of the retrial rate in M/M/l and M/G/l queues, respectively, with exponential retrials, introducing for this purpose a new Markovian description. Warfield and Foers [7,8] use a Bayesian approach to estimate the traffic intensity for different models of Teletraffic in a Markovian context where primary and repeated flows are subsumed into a homogeneous Poisson process with a different arrival rate. Other studies that allow for retrials have been done by Hoffman and Harris [9] and Lewis and Leonard [lo]. In this article, we deal with some statistical aspects in the estimation of unknown parameters for the less restrictive G/G/l queue, with general retrials, where the time before a customer tries his call again does not have to be exponentially distributed. We begin with the case of a nonconstant Ilow of retrials where all customers retry in order to get service. We now describe the model. This research was supported PB95-0416.
by the European
089571771991% - see front matter. PII:SO895-7177(99)00142-9
Commission
through INTAS Grant 960828
@ 1999 Elsevier Science Ltd. All rights reserved.
and DGICYT
Typeset
by &+Q$
Grant
Large Sample Inference
In this paper we use the likelihood 0, I$, and 77 when we consider r(T),
“marked” that
usually
general
available,
customers
are not marked
4, we consider
for G/G/l
In the next
we assume
to G/G/l
queues.
3 with the estimation estimators
section
we can observe
retrial
queues
Since complete of the mean
we cannot
Gaussian a retrial
distributed.
of the parameters
{wk; 1 5 k 5 T}, i.e., each customer
and in consequence
and asymptotically
in Section
estimators
where
his call. We extend
[12,13] obtained
we deal in Section
we get consistent Finally,
information)
< t 5 T} and also the vector and Prabhu
exponentially
case (complete
and we know when he repeats
Basawa
(1.2) to get consistent
f, g, and h are not assumed
the most
{X(t);0
function
199
in orbit
is
the results
information
is not
interretrial
time when
tell who makes a retrial.
In this case
even when {X(t); t 2 0) is nonergodic.
queue with a constant
flow of repeated
attempts.
2. MAXIMUM LIKELIHOOD ESTIMATORS UNDER COMPLETE INFORMATION In this section be observed. Our analysis
we assume
that
all the variables
involved
in the likelihood
function
(1.1) can
When there are no retrials, (1.1) becomes the likelihood function given in [13]. is similar to theirs but we allow for retrials. In a classical G/G/l queue, when the
conditions - F(zL’; 0)) 5
0,
T T 00, a.s.,
(cl)
log(1 - G(s*; 4)) 5
0,
T T cm, a.s.,
(c2)
n-i~2&log(l m-ij2$ hold (where p denotes equivalent
in the sense that expression that
convergence
in probability),
the likelihood
function
(1 .l) is asymptotically
to
the maximum
likelihood
estimators
above, have the same limit distribution
of 0 and 4 obtained
using
(1.1) and the
(see [12,13]). In the same way, it can be proved
if we add the condition log(1 - H(&q))
the likelihood
function
(1 .l) and (1.2) are equivalent.
2
0,
T T co a.s.,
Then, we can consider
(c3)
the simpler
likelihood
function (1.2) instead of (1.1). Basawa and Prabhu [13) prove that conditions (cl) and (~2) are satisfied by a large class of distributions F and G even when the queue is nonergodic. However, in general, retrial queues condition (~3) might not hold when the stochastic process {X(t); t > 0) is nonergodic. We will show this later in Remark 3.2. If {X(t);t 2 0) is ergodic, i.e., if there exists an asymptotic stationary distribution, and T is fixed, w;, 1 5 k 5 X(T), h as a proper distribution, and hence condition
(~3) is satisfied
when (6)
log(1 - H(wi;
q)) is a continuous
is also satisfied if T is the instant when the mth departure distribution, with density given by h(w;q)
=
0 g 7)
p
wP-l
~
(p_l)!exp
function.
occurs and retrials
{-7 >.
We can prove that
k=l
Condition
(~3)
have an Erlangian
A. F~ODRIGO AND
200
m 1 0) is ergodic,
If {X(T)
E X(m);
condition
(~3) is satisfied.
If n, m, and r are nonrandom, densities
V.~ZQUEZ
M.
w; + ... + wkCrn) < co, a.s., when m T 00, and hence
it is well known
f, g, and h (see [14)), the maximum
that
under some regularity
likelihood
(ML) estimators,
conditions
on the
0, 6, and 7j, of the
parameters 8, C#J,and 17 exist, are consistent and asymptotically Gaussian (when n, m, and T tend to infinity). In our case the variables n, m, and T depend on the stopping time T and are therefore
random
and mutually
we can fix one of them. to infinity
as T T 00, a.s.
ML estimators
dependent.
However, Hence,
Hence, we cannot
we have assumed
that
we can still apply
of the parameters
fix them simultaneously;
these random
variables
the law of large numbers
0, 4, and n. Moreover,
if the stability
at most
increase,
a.s.,
to get consistent
conditions
on stopping
times E[n(T)) n(T)
L
i
E]m(T)]
P_
i
49
’
J?-+(~)] r(T)
’
are satisfied we could prove, in a similar ML estimators are asymptotically Gaussian
5
1 ’
manner as Basawa and Prabhu and mutually independent.
To obtain straightforward results we assume that the stopping when the mth departure occurs (although we would get analogous provided
the stability
conditions
as T T 00 as.,
in (2.1) were satisfied).
(2.1)
(131 did,
that
the
time T z T(m) is the instant results for other stopping rules
In what follows we assume that a G/G/l
retrial queue is observed over the time interval (O,T(m)] and that conditions (cl) and (~3) hold (note that since T E T(m), condition (~2) is not necessary). In this case we have the following asymptotic LEMMA
results.
2.1.
Let n
be the number of primary arrivals occurring before the mth departure. Then,
for any general retrial distribution,
where p = E[s]/E[U] is the traffic intensity. PROOF.
Lot X(m)
be the number
of customers
We can then write n/m = 1 +X(m)m-‘.
in the orbit right after the mth departure
If &,,+.&?[X(m)]
< co, then X(m)m-’
occurs.
-% 0, when
m T 00, and hence n/m -% 1, m T co. On the other hand we can write
n
-=-m
n(T) 1’ T
(2.2)
m’
where n(t), t 2 0, is a renewal process with renewals occurring at the arrival times. a.s., when m T 00, it follows from the elementary renewal theorem that
n(T) -+ (E[u])-‘,
7
The term T/m can be written
Since T T 00,
a.s.
(2.3)
as
(2.4)
where Ij is the idle period after the jth departure. By the strong law of large numbers member in (2.4) tends, a.s., to E[s] as m 1 00. Let X and I be such that Pr[X = z] = dl-rPr[X(m)
= z]
and
Pr[l < y] = jl~mirPr[lj
5 Y].
the first
Large Sample Inference = 00 and X = 00 with probability
If lim,,,E[X(m)]
2
from (2.2), (2.3), and (2.4), n/m E[lj]
201
one, I = 0 with probability
p > 1, m t 00, (note that n/m
2 1 always).
one.
Then,
Note ako that
5 E[u], and hence
Then,
= 0;) with Pr[X
if 6,,,E[X(m)]
and therefore
n/m
5
< co] > 0, we have that
lim,,,E[X(m)/m]
= 0
1, m 7 CXJ.
I
LEMMA 2.2. Let T be the total number
of retrials
{X(t);
retrid
t 2 0) is ergodic,
for any general
r
occurring
ELWbl Eb-4 ’
P,
m
before the mth departure.
Then,
if
distribution
m t 00,
where X = limt_,oo X(t). PROOF. Since T t 00 a.s. as m t 00, if {X(t); t 2 0) is ergodic,
the ergodic
theorem
guarantees
that 1
=
X(t) dt 5
E[X],
m too.
(2.5)
X(m) 1 w;.
(2.6)
Kl J Note also that 1 = ; o X(t)dt=;f:w,,+; J k=l
k=l
side in (2.6) tends to zero in Since X(m) is finite a.s., the second term on the right-hand probability when m 7 00. Then, using the strong law of large numbers, the expression (2.6) tends that, r/m
in probability
to E(w].
in the ergodic
case, p < 1, and hence T/m
-% E[XJE[u]/E[w],
THEOREM 2.1. Suppose conditions
Hence r/T
2
E[X]/E[
w ], m 1 00. From Lemma
= (T/n)(n/m)
L
2.1 it follows
E[u], m 1 00. Therefore,
m t co.
I
that the density
(see (141). Then,
functions
f, g, and h satisfy
the appropriate
regularity
when m 1 00, rn1i2
(
8 - 0
>
+ N (0, ~-‘a~)
,
(2.7)
and
+N(&(T$),
m1/2($-@)
(2.8)
where a = max(1, p),
(+) denotes ergodic
convergence
in distribution
and
N is a normal
m’j2(7j - 7) + N(o, (E[u]E[X])-‘E(w]a$, where g2 = 11
Moreover, the random independent .
variables
distribution. m t 00,
If {X(t);
t 2 0) is (2.9)
(E[(~log(h(wk:n)))2]}-1.
m’i2(8
- 0), m1i2($
- 4), and m1i2(fj - 11) are asymptotically
A.
202
PROOF.
From the central
RODRIGO
limit theorem
AND
for random
M. V~ZQUEZ sums of iid random
variables
(see 115, p. 143])
we have that n-1/2
mT
00,
i=l
and hence likelihood
provided
from Lemma theory.
2.1 we get (2.7).
The result
in (2.8) follows directly
from maximum
To get (2.9), note that
that we can use the central
limit theorem
for the sum of T (random)
iid random
variables.
Hence, r”2(?j By changing
the normalizer
m 7 03.
- 7)) =S N (0, u,“) ,
T to m, we obtain
(2.9) from Lemma
2.2.
Finally,
the asymptotic
independence of the three estimators can be easily proved, as Basawa and Prabhu 1121 did for classical G/G/l queues, using the Cramer-Wold device together with Lemmas 2.1 and 2.2. 1
3. ESTIMATION
OF THE MEAN
INTERRETRIAL
TIME
In this section we focus on the estimation of the mean interretrial time, E[w] = 11. As in the previous section, we observe a G/G/l retrial queue over the time interval (O,T], T E T(m). To estimate T,Jwe could use the sample mean
(3.1) k=l
but the interretrial times wk are not usually available. However, from equation (2.6), we can use the process X(t) to get an estimator of 77. We state this in the following proposition. PROPOSITION 3.1.
If {X(t);
t 2 0) is ergodic,
fj* =
the estimators
6 and
TX(t)& 1 7. J 0
are consistent
and asymptotically
equivalent, i.e., r’j2(7j - T]) and r112(ij* - 77) are both N(O,
Var(w))
as m T co.
PROOF. equation
The consistency part follows from the law of large numbers can be written as
and equation
(2.6).
This
X(m)
,1/Q* -
17) =
g/2(< -
Q) +
7.-w
c
w;.
k=l
Using the central limit theorem for random sums r 1/2(q-Q) * N(O,Var(ur)). Since {X(t);t 2 0) is ergodic the second term on the right-hand side tends to zero in probability, when m T co and the proposition follows. I REMARK 3.1. Note that in evaluating ?j* we do not need to know the interretrial times, that is, we do not need to mark the customers in orbit, but we still need to observe the process {X(t); t 2 0) which might not always be available. The telephone network is a straightforward example of a retrial system although it seems to be quite difficult to observe the orbit and to distinguish between primary or repeated calls. However, we can do the following. Consider one agent serving two telephone lines A and B. The phone number of A (line of primary customers)
Large Sample Inference
is advertised
and hence that
an answering
machine
which then both
becomes
line takes all primary
asks this customer
call counter available
in both
We now obtain Proposition
Then,
service.
arrives,
he starts
with this scheme,
and the server is busy,
of line B (line of repeated
Hence, all calls to B are repeated his service.
Assume
that
calls) calls.
If
the server has a
the server has all the relevant
information
4’.
a result
for the nonergodic
case which
3.1. To do this, we first prove the following
LEMMA 3.1. PROOF.
lines.
for evaluating
If a call occurs
to phone the number
the only way of getting
lines are free when a customer
calls.
203
Pr[X(m)
Iflim,,,
is analogous
to the result
< k] = 0, Vk 2 0, then 3c > 0 such that r/m2
We can write
given
in
lemma. -% c.
m-1 E[r]
=
c
E[W)I,
i=o
For a fixed i, where AT-(~) is the number of retrials between the ith and the (i + l)th departures. E(Ar(i)] = p(i) + E[Ar*(i)], where p(i) is the probability that the next customer getting service corresponds to a repeated call and Ar*(i) is the number of retrials occurring while he is being served. Then, if there are CC*customers in orbit when the ith service starts, AT*(~) is the number of renewals during mean interrenewal Lemma
a service of a number Z* of delayed and independent renewal processes with times equal to 77. Then, for i large enough, E[Ar(i)] = O(E[X(i)]). From 2.1, E[X(i)] = O(i). H ence, E[r] = 0(m2) and therefore r/m2 tends to a constant c,
c > 0.
I
PROPOSITION
3.2.
the assumption given in Lemma
Under
3.1, the estimators ij and 7j* remain
consistent and, when m T co, rnli2(7j - 7j) -5 PROOF.
From the law of large numbers m
rj is consistent.
Yg ( w;=
m1j2(7j* - 77) 5
0,
X(m) -_ m
m r
)(
0.
We can now write 1 X(m)
x(m) kzl wi 4
(3.3)
’
where X(m)/m is bounded (see proof of Lemma 2.1) and m/r tends to zero in probability. Then, the first bracket on the right-hand side of (3.3) tends to zero in probability as m T co, while the term in the second bracket
is finite a.s. Hence Q* is a consistent
estimator
for the mean interretrial
time 7. From (2.6) we get
m1/2(fj* - 77)=
From
the central
together
limit
with Lemma
($)
theorems
r1j2(ij - q) + (FF)
for random
sums,
(&EJw;).
r1i2(fi - 77) + N(O,Var(w)),
(3.4)
m T co and,
3.1, we get m1i2(lj - 77) -% 0, m 1 00, and hence from (3.4) m1i2(ij* - 77)
-% 0, m T 00.
a
REMARK 3.2. The asymptotic equivalence we have just established is not an equivalence in the sense of Proposition 3.1. Using r1j2 as a normalizer, when limm--rm Pr(X(m) < k] = 0, V k 1 0, from Lemma 3.1 and equality (2.6), we get r112(fi* - n) =+ N(c,Var(w)),
mfcq
c>O,
but G2(7j
- v) * N(0, Var(w)),
m f co.
A. RODRICO
204 For an M/G/l
queue with retrials
exp(l/n)
AND
M.
VliZQUEZ
we have
bWm)l)1/2(ir* - d * JV([27-Pi% - l)] v2v, v2), Note that
when p = 1 we also have an asymptotic
for general
retrial
Pr[X(m)
when lim,,, REMARK
queues
3.3. Another
Since the variables
by observing
equivalence.
Analogous
E[X(m)]{E[r(m)]}-1/2
that
m T co.
tends
results
are obtained
in probability
to zero
< co] > 0. way to estimate
involved
77 in the ergodic
1
2=-
T
T
J X(t) o
could be the best alternative. In M/G/l retrial for E(X] from which we obtain the estimator
fjl
=
case, is via any estimator
of E[X] are not iid, the integral
in the estimation
queues
f of E[X].
estimator
4 we have (see [3]) an explicit
expression
2(1- P)i - Pp2 ’
2xp
expression for the variance of $1 where E[u] = l/X and Pz = E[s2]. W e can get an asymptotic as Rodrigo et al. [6] did for the case when we cannot distinguish between retrials and primary arrivals. REMARK 3.4. In M/G/l observable characteristics
ergodic retrials queues, with exponential retrials, we can use other of the system to estimate n such as the number of calls (primary and/or
repeated) during a service, the length of the busy periods, the number of services occurring in a busy period, the time spent in the system by each customer, etc. The mean values of these characteristics
in steady
state
are known
(see [3,16]) and from them we can obtain
estimators
of n as we did in the previous remark. For instance, if we only observe the number of arrivals during each service and we cannot not distinguish between primary and repeated arrivals, we can estimate the retrial rate p = 71-l using the estimator b=
2(1- P)4- 2P %32
)
where
and Qi, 1 5 i 5 m, is the number
of arrivals
4. CONSTANT
during
the ith service.
FLOW OF REPEATED
CALLS
In the retrial systems studied till now we have assumed that all customers in orbit compete to obtain service, that is, each customer tries his luck again until he finds the service area free and he does this independently of how other customers in orbit behave. We now consider the case when retrials from the orbit occur in an ordered manner resulting in an output flow of repeated calls which is independent of the number of items in the orbit. These retrial systems were introduced by Fayolle [l] and Farahmand (21. We next do an analysis similar to that of Section 2. We assume that conditions (cl) and (~2) hold. Condition (~3) now becomes r-1/2
6
log(1 - H(w’; II)))
JL
0,
T T co a.s.,
(c3)
Large Sample Inference
if T = t is nonrandom.
which is satisfied occurs, {X(t);
this condition
is also satisfied
t 2 0) is nonergodic.
If T = T(m) is the instant
As in Section
For any general
retrial
2, we assume
distribution
-n -5 m
PROOF.
Proceeding
process.
For the nonergodic
that
(cl)
LEMMA 4.2.
max(1, p’),
even when
queue is observed
=mfcQ idle period after esch departure.
as we did in Lemma 2.1 we conclude
Let {X(t);
retrial
and (~3) hold. In this case we have the
that n/m
case, when m is large enough,
5
1 when X(t) is an ergodic
the idle periods
distributed as min(w 1 w > w*,u 1 u > u’) since the orbit Using (2.2), (2.3), and (2.4), the lemma is proved.
distribution
distribution,
a G/G/l
we have that
and I is the stationary
where p* = p + (E[u])-‘E[I]
when the mth departure
at least when H is an Erlangian
over the time interval (O,T(m)] and that conditions asymptotic results given below. LEMMA 4.1.
205
t 2 0) be an ergodic stochastic
is occupied
after departures with
are
probability
one. I
process.
Then,
for any general retrial
we have $ 5
Moreover, when {X(t);
(E[w])-’
Pr[X
2 11,
m t 00.
t > 0) is nonergodic,
$2 Let To and Tl be the amount respectively. We can write
PROOF.
m t 00.
(E[w])-‘,
of time in (0, T] during
which the orbit is empty
and busy,
T _- r/T1 T - T/Tl’ where Tl = w* + WI + .e. + w,. and w* < 00, a.s. Then, r/T1 tends in probability On the other hand, T/T1 tends in probability to Pr[X 2 11, if the system is ergodic,
to E[w]-l. and to one
otherwise.
I
THEOREM 4.1.
under
Let 6, 4, and 6 be the ML estimators
the appropriate
ML estimators
regularity
conditions
6, d, and +j are consistent rnli2(8
obtained
on the density
with asymptotic
- 0) =+ N (0, a_la,2)
,
from likelihood
functions
(1.2).
Then,
f, g, and h (see [141), the
distributions
m1i2($
- 4) +
N (0, c$) ,
and m’i2(ij where a = max(l,p’) x(E[w])-’ PROOF.
and < = E[u](E[w])-1
in the nonergodic
asymptotically
case.
N (~,a:<-‘)
- 77) +
Moreover,
2 l] in the ergodic case, and < = p’E[u] ^ m’i2(0 - 0), m’/2(q4 - 4), and m’i2(7j - r,~)are
Pr[X
independent.
The theorem
and 4.2 instead REMARK 4.1.
,
can be proved proceeding of Lemmas 2.1 and 2.2.
If the densities
as we did in Theorem distributed
-n -%max(l,p+X(X+~)-‘) m pmin(l,Pr[X
with parameters
=a,
2 11) = pb,
m’i2(+j - 7) =S N(o, c), where c = Xp(ab)-‘.
Note that p + X(X + II)-’
4.1 a
f and h are exponentially
& 2
2.1 but using Lemmas
2 1 if X(t)
is nonergodic.
X and ~1,
A. RODRIGO AND M. VP;ZQUEZ
206 REMARK
The case of constant
4.2.
who find the server
retrials
busy wait in a “room”
can be interpreted and the server
in another must
way. The customers
search
for customers
after
a
random idle period which is independent of the number of waiting customers. Note that in this case there are no retrials when the service is busy and therefore it is not exactly the same as the system
just described.
can also consider
priority
If we know the number between
primary
the parameter customers
of customers
and repeated
in orbit
arrivals,
77. For instance, arrivals
are called
“queues
on the number
with priority of customers
at the end of each service
we can easily
if the /ct” idle period
primary
and yk is the duration
and repeated
of q would be the unique
where i = (m--1+x,)/T. estimators
which depend
obtain,
in both
is exp(zcr,/q),
arrivals
solution
of the kth idle period.
and primary
arrivals
can also be used to estimate
the parameter
property
We
(room).
and we can distinguish cases,
an estimator
for of
where r is the number
If we could not distinguish
occur at rate A, the ML estimator
(which exists with probability
Because of the memoryless
intervals”. in orbit
where zk is the number
in orbit at the end of the kth service, we get the ML estimator
of nonprimary between
In [17] these systems intervals
one) of the likelihood
of exponential
7 in the standard
M/G/l
equation
distributions, retrial
these
queue (see
Section 2) with interretrial times exp(l/q). Note that although we need less sample information to get these estimators we cannot use whatever information we might have on unsuccessful retrials.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
G. Fayolle, A simple telephone exchange with delayed feedbacks, In Teletmfic Analysis Computer, pp. 245-253, Elsevier Science, (1986). K. Farahmand, Single line queue with repeated demands, Queueing Systems 6, 223-228 (1990). G.I. Falin, A survey of retrial queues, Queuing Systems 7, 127-168 (1990). G.I. Falin and J.G.C. Templeton, Retrial Queues, Chapman and Hall, (1997). G.I. Falin, Estimation of retrial rate in a retrial queue, Queueing Systems 19, 231-246 (1995). A. Rodrigo, M. Vazques and G.I. Falin, A new Markovian description of the M/G/I retrial queue, European Journal of Operational Research 104, 231-240 (1998). R.E. Warfield and G.A. Foers, Application of Bayesian methods to teletrafiic measurement and dimensioning, Austmlian Telecommunications Research 18, 51-58 (1984). R.E. Warfield and G.A. Foers, Application Bayesian teletraflic measurement to systems with queueing or repeated attempts, Proceedings of the Eleventh International Teletmfic Congress (1985). K.L. Hoffman and C.M. Harris, Estimation of a caller retrial rate for a telephone information system, European Journal of Opemtional Research 27, 207-214 (1986). A. Lewis and G. Leonard, Measurements of repeat call attempts in the intercontinental telephone service, Proceedings of the Tenth International Teletmfic Congress (1982). P. Billingsley, Statistical Inference for Markov Processes, The University of Chicago Press, (1961). I.V. Basawa and N.U. Prabhu, Estimation in single server queues, Naval Research Logistics Quarterly 28, 475487 (1981). I.V. Basawa and N.U. Prabhu, Large sample inference from simple server queues, Queueing Systems 1, 289-304 (1988). C.R. Rae, Linear Statistical Inference and Its Applications, 2”d edition, John Wiley, (1973). P. Billingsley, Convergence of Probability Measures, John Wiley, (1968). G.I. Falin and C. Fricker, On the virtual waiting time in an M/G/l retrial queue, Journal of Applied P&ability 28, 446460 (1991). M. Martin and J.R. Artalejo, Analysis of an M/G/l queue with two types of impatient units, Advances in Applied P&ability 27, 84G861 (1995).
Mathematical and Computer Modelling30 (1999)
197-206
www.elsevier.nl/locate/mcm
Large Sample Inference in Retrial Queues A. RODRIGO AND M. VAZQUEZ Departamento de Anaiisis Economico Facultad de Ciencias Economicas Universidad Complutense de Madrid Campus de Somosaguas, 28223 Madrid, Spain ececo03Qeis.ucm.es Abstract-we
analyze retrial queueing systems from a statistical viewpoint. We consider a
general G/G/l retrial queue where the flow of repeated attempts can be non-Markovian. We observe the operation of the system over a random time interval (O,T] focusing on the case where T is a departure epoch. We obtain some asymptotic relationship among the random variables involved, This allows us to get asymptotically Gaussianconsistent even when the system in nonergodic. estimators for the unknown parameters in a parametric context. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-G/G/l
retrial queue, Maximum likelihood estimation,
General retrials.
1. INTRODUCTION In communication networks customers do not usually wait in a queue when the service area is busy. They leave and try again some time later. We say that they join a new flow of repeated arrivals (calls) which is different from the primary flow. More precisely, if a customer finds the service area busy he “goes to an orbit” wherefrom he retries his call until he finds the service free. If the service area is free the customer starts to be served immediately, leaving the system after he has finished his service. The flow of repeated calls can be constant (see [1,2]) or depend on the number of customers in orbit (see the surveys by Falin [3,4]). Many articles have been published dealing with the theory of retrial systems but very few analyze these systems from a statistical viewpoint. Falin [5] and Rodrigo et al. 16) construct integral estimators of the retrial rate in M/M/l and M/G/l queues, respectively, with exponential retrials, introducing for this purpose a new Markovian description. Warfield and Foers [7,8] use a Bayesian approach to estimate the traffic intensity for different models of Teletraffic in a Markovian context where primary and repeated flows are subsumed into a homogeneous Poisson process with a different arrival rate. Other studies that allow for retrials have been done by Hoffman and Harris [9] and Lewis and Leonard [lo]. In this article, we deal with some statistical aspects in the estimation of unknown parameters for the less restrictive G/G/l queue, with general retrials, where the time before a customer tries his call again does not have to be exponentially distributed. We begin with the case of a nonconstant Ilow of retrials where all customers retry in order to get service. We now describe the model. This research was supported PB95-0416.
by the European
089571771991% - see front matter. PII:SO895-7177(99)00142-9
Commission
through INTAS Grant 960828
@ 1999 Elsevier Science Ltd. All rights reserved.
and DGICYT
Typeset
by &+Q$
Grant
Large Sample Inference
In this paper we use the likelihood 0, I$, and 77 when we consider r(T),
“marked” that
usually
general
available,
customers
are not marked
4, we consider
for G/G/l
In the next
we assume
to G/G/l
queues.
3 with the estimation estimators
section
we can observe
retrial
queues
Since complete of the mean
we cannot
Gaussian a retrial
distributed.
of the parameters
{wk; 1 5 k 5 T}, i.e., each customer
and in consequence
and asymptotically
in Section
estimators
where
his call. We extend
[12,13] obtained
we deal in Section
we get consistent Finally,
information)
< t 5 T} and also the vector and Prabhu
exponentially
case (complete
and we know when he repeats
Basawa
(1.2) to get consistent
f, g, and h are not assumed
the most
{X(t);0
function
199
in orbit
is
the results
information
is not
interretrial
time when
tell who makes a retrial.
In this case
even when {X(t); t 2 0) is nonergodic.
queue with a constant
flow of repeated
attempts.
2. MAXIMUM LIKELIHOOD ESTIMATORS UNDER COMPLETE INFORMATION In this section be observed. Our analysis
we assume
that
all the variables
involved
in the likelihood
function
(1.1) can
When there are no retrials, (1.1) becomes the likelihood function given in [13]. is similar to theirs but we allow for retrials. In a classical G/G/l queue, when the
conditions - F(zL’; 0)) 5
0,
T T 00, a.s.,
(cl)
log(1 - G(s*; 4)) 5
0,
T T cm, a.s.,
(c2)
n-i~2&log(l m-ij2$ hold (where p denotes equivalent
in the sense that expression that
convergence
in probability),
the likelihood
function
(1 .l) is asymptotically
to
the maximum
likelihood
estimators
above, have the same limit distribution
of 0 and 4 obtained
using
(1.1) and the
(see [12,13]). In the same way, it can be proved
if we add the condition log(1 - H(&q))
the likelihood
function
(1 .l) and (1.2) are equivalent.
2
0,
T T co a.s.,
Then, we can consider
(c3)
the simpler
likelihood
function (1.2) instead of (1.1). Basawa and Prabhu [13) prove that conditions (cl) and (~2) are satisfied by a large class of distributions F and G even when the queue is nonergodic. However, in general, retrial queues condition (~3) might not hold when the stochastic process {X(t); t > 0) is nonergodic. We will show this later in Remark 3.2. If {X(t);t 2 0) is ergodic, i.e., if there exists an asymptotic stationary distribution, and T is fixed, w;, 1 5 k 5 X(T), h as a proper distribution, and hence condition
(~3) is satisfied
when (6)
log(1 - H(wi;
q)) is a continuous
is also satisfied if T is the instant when the mth departure distribution, with density given by h(w;q)
=
0 g 7)
p
wP-l
~
(p_l)!exp
function.
occurs and retrials
{-7 >.
We can prove that
k=l
Condition
(~3)
have an Erlangian
A. F~ODRIGO AND
200
m 1 0) is ergodic,
If {X(T)
E X(m);
condition
(~3) is satisfied.
If n, m, and r are nonrandom, densities
V.~ZQUEZ
M.
w; + ... + wkCrn) < co, a.s., when m T 00, and hence
it is well known
f, g, and h (see [14)), the maximum
that
under some regularity
likelihood
(ML) estimators,
conditions
on the
0, 6, and 7j, of the
parameters 8, C#J,and 17 exist, are consistent and asymptotically Gaussian (when n, m, and T tend to infinity). In our case the variables n, m, and T depend on the stopping time T and are therefore
random
and mutually
we can fix one of them. to infinity
as T T 00, a.s.
ML estimators
dependent.
However, Hence,
Hence, we cannot
we have assumed
that
we can still apply
of the parameters
fix them simultaneously;
these random
variables
the law of large numbers
0, 4, and n. Moreover,
if the stability
at most
increase,
a.s.,
to get consistent
conditions
on stopping
times E[n(T)) n(T)
L
i
E]m(T)]
P_
i
49
’
J?-+(~)] r(T)
’
are satisfied we could prove, in a similar ML estimators are asymptotically Gaussian
5
1 ’
manner as Basawa and Prabhu and mutually independent.
To obtain straightforward results we assume that the stopping when the mth departure occurs (although we would get analogous provided
the stability
conditions
as T T 00 as.,
in (2.1) were satisfied).
(2.1)
(131 did,
that
the
time T z T(m) is the instant results for other stopping rules
In what follows we assume that a G/G/l
retrial queue is observed over the time interval (O,T(m)] and that conditions (cl) and (~3) hold (note that since T E T(m), condition (~2) is not necessary). In this case we have the following asymptotic LEMMA
results.
2.1.
Let n
be the number of primary arrivals occurring before the mth departure. Then,
for any general retrial distribution,
where p = E[s]/E[U] is the traffic intensity. PROOF.
Lot X(m)
be the number
of customers
We can then write n/m = 1 +X(m)m-‘.
in the orbit right after the mth departure
If &,,+.&?[X(m)]
< co, then X(m)m-’
occurs.
-% 0, when
m T 00, and hence n/m -% 1, m T co. On the other hand we can write
n
-=-m
n(T) 1’ T
(2.2)
m’
where n(t), t 2 0, is a renewal process with renewals occurring at the arrival times. a.s., when m T 00, it follows from the elementary renewal theorem that
n(T) -+ (E[u])-‘,
7
The term T/m can be written
Since T T 00,
a.s.
(2.3)
as
(2.4)
where Ij is the idle period after the jth departure. By the strong law of large numbers member in (2.4) tends, a.s., to E[s] as m 1 00. Let X and I be such that Pr[X = z] = dl-rPr[X(m)
= z]
and
Pr[l < y] = jl~mirPr[lj
5 Y].
the first
Large Sample Inference = 00 and X = 00 with probability
If lim,,,E[X(m)]
2
from (2.2), (2.3), and (2.4), n/m E[lj]
201
one, I = 0 with probability
p > 1, m t 00, (note that n/m
2 1 always).
one.
Then,
Note ako that
5 E[u], and hence
Then,
= 0;) with Pr[X
if 6,,,E[X(m)]
and therefore
n/m
5
< co] > 0, we have that
lim,,,E[X(m)/m]
= 0
1, m 7 CXJ.
I
LEMMA 2.2. Let T be the total number
of retrials
{X(t);
retrid
t 2 0) is ergodic,
for any general
r
occurring
ELWbl Eb-4 ’
P,
m
before the mth departure.
Then,
if
distribution
m t 00,
where X = limt_,oo X(t). PROOF. Since T t 00 a.s. as m t 00, if {X(t); t 2 0) is ergodic,
the ergodic
theorem
guarantees
that 1
=
X(t) dt 5
E[X],
m too.
(2.5)
X(m) 1 w;.
(2.6)
Kl J Note also that 1 = ; o X(t)dt=;f:w,,+; J k=l
k=l
side in (2.6) tends to zero in Since X(m) is finite a.s., the second term on the right-hand probability when m 7 00. Then, using the strong law of large numbers, the expression (2.6) tends that, r/m
in probability
to E(w].
in the ergodic
case, p < 1, and hence T/m
-% E[XJE[u]/E[w],
THEOREM 2.1. Suppose conditions
Hence r/T
2
E[X]/E[
w ], m 1 00. From Lemma
= (T/n)(n/m)
L
2.1 it follows
E[u], m 1 00. Therefore,
m t co.
I
that the density
(see (141). Then,
functions
f, g, and h satisfy
the appropriate
regularity
when m 1 00, rn1i2
(
8 - 0
>
+ N (0, ~-‘a~)
,
(2.7)
and
+N(&(T$),
m1/2($-@)
(2.8)
where a = max(1, p),
(+) denotes ergodic
convergence
in distribution
and
N is a normal
m’j2(7j - 7) + N(o, (E[u]E[X])-‘E(w]a$, where g2 = 11
Moreover, the random independent .
variables
distribution. m t 00,
If {X(t);
t 2 0) is (2.9)
(E[(~log(h(wk:n)))2]}-1.
m’i2(8
- 0), m1i2($
- 4), and m1i2(fj - 11) are asymptotically
A.
202
PROOF.
From the central
RODRIGO
limit theorem
AND
for random
M. V~ZQUEZ sums of iid random
variables
(see 115, p. 143])
we have that n-1/2
mT
00,
i=l
and hence likelihood
provided
from Lemma theory.
2.1 we get (2.7).
The result
in (2.8) follows directly
from maximum
To get (2.9), note that
that we can use the central
limit theorem
for the sum of T (random)
iid random
variables.
Hence, r”2(?j By changing
the normalizer
m 7 03.
- 7)) =S N (0, u,“) ,
T to m, we obtain
(2.9) from Lemma
2.2.
Finally,
the asymptotic
independence of the three estimators can be easily proved, as Basawa and Prabhu 1121 did for classical G/G/l queues, using the Cramer-Wold device together with Lemmas 2.1 and 2.2. 1
3. ESTIMATION
OF THE MEAN
INTERRETRIAL
TIME
In this section we focus on the estimation of the mean interretrial time, E[w] = 11. As in the previous section, we observe a G/G/l retrial queue over the time interval (O,T], T E T(m). To estimate T,Jwe could use the sample mean
(3.1) k=l
but the interretrial times wk are not usually available. However, from equation (2.6), we can use the process X(t) to get an estimator of 77. We state this in the following proposition. PROPOSITION 3.1.
If {X(t);
t 2 0) is ergodic,
fj* =
the estimators
6 and
TX(t)& 1 7. J 0
are consistent
and asymptotically
equivalent, i.e., r’j2(7j - T]) and r112(ij* - 77) are both N(O,
Var(w))
as m T co.
PROOF. equation
The consistency part follows from the law of large numbers can be written as
and equation
(2.6).
This
X(m)
,1/Q* -
17) =
g/2(< -
Q) +
7.-w
c
w;.
k=l
Using the central limit theorem for random sums r 1/2(q-Q) * N(O,Var(ur)). Since {X(t);t 2 0) is ergodic the second term on the right-hand side tends to zero in probability, when m T co and the proposition follows. I REMARK 3.1. Note that in evaluating ?j* we do not need to know the interretrial times, that is, we do not need to mark the customers in orbit, but we still need to observe the process {X(t); t 2 0) which might not always be available. The telephone network is a straightforward example of a retrial system although it seems to be quite difficult to observe the orbit and to distinguish between primary or repeated calls. However, we can do the following. Consider one agent serving two telephone lines A and B. The phone number of A (line of primary customers)
Large Sample Inference
is advertised
and hence that
an answering
machine
which then both
becomes
line takes all primary
asks this customer
call counter available
in both
We now obtain Proposition
Then,
service.
arrives,
he starts
with this scheme,
and the server is busy,
of line B (line of repeated
Hence, all calls to B are repeated his service.
Assume
that
calls) calls.
If
the server has a
the server has all the relevant
information
4’.
a result
for the nonergodic
case which
3.1. To do this, we first prove the following
LEMMA 3.1. PROOF.
lines.
for evaluating
If a call occurs
to phone the number
the only way of getting
lines are free when a customer
calls.
203
Pr[X(m)
Iflim,,,
is analogous
to the result
< k] = 0, Vk 2 0, then 3c > 0 such that r/m2
We can write
given
in
lemma. -% c.
m-1 E[r]
=
c
E[W)I,
i=o
For a fixed i, where AT-(~) is the number of retrials between the ith and the (i + l)th departures. E(Ar(i)] = p(i) + E[Ar*(i)], where p(i) is the probability that the next customer getting service corresponds to a repeated call and Ar*(i) is the number of retrials occurring while he is being served. Then, if there are CC*customers in orbit when the ith service starts, AT*(~) is the number of renewals during mean interrenewal Lemma
a service of a number Z* of delayed and independent renewal processes with times equal to 77. Then, for i large enough, E[Ar(i)] = O(E[X(i)]). From 2.1, E[X(i)] = O(i). H ence, E[r] = 0(m2) and therefore r/m2 tends to a constant c,
c > 0.
I
PROPOSITION
3.2.
the assumption given in Lemma
Under
3.1, the estimators ij and 7j* remain
consistent and, when m T co, rnli2(7j - 7j) -5 PROOF.
From the law of large numbers m
rj is consistent.
Yg ( w;=
m1j2(7j* - 77) 5
0,
X(m) -_ m
m r
)(
0.
We can now write 1 X(m)
x(m) kzl wi 4
(3.3)
’
where X(m)/m is bounded (see proof of Lemma 2.1) and m/r tends to zero in probability. Then, the first bracket on the right-hand side of (3.3) tends to zero in probability as m T co, while the term in the second bracket
is finite a.s. Hence Q* is a consistent
estimator
for the mean interretrial
time 7. From (2.6) we get
m1/2(fj* - 77)=
From
the central
together
limit
with Lemma
($)
theorems
r1j2(ij - q) + (FF)
for random
sums,
(&EJw;).
r1i2(fi - 77) + N(O,Var(w)),
(3.4)
m T co and,
3.1, we get m1i2(lj - 77) -% 0, m 1 00, and hence from (3.4) m1i2(ij* - 77)
-% 0, m T 00.
a
REMARK 3.2. The asymptotic equivalence we have just established is not an equivalence in the sense of Proposition 3.1. Using r1j2 as a normalizer, when limm--rm Pr(X(m) < k] = 0, V k 1 0, from Lemma 3.1 and equality (2.6), we get r112(fi* - n) =+ N(c,Var(w)),
mfcq
c>O,
but G2(7j
- v) * N(0, Var(w)),
m f co.
A. RODRICO
204 For an M/G/l
queue with retrials
exp(l/n)
AND
M.
VliZQUEZ
we have
bWm)l)1/2(ir* - d * JV([27-Pi% - l)] v2v, v2), Note that
when p = 1 we also have an asymptotic
for general
retrial
Pr[X(m)
when lim,,, REMARK
queues
3.3. Another
Since the variables
by observing
equivalence.
Analogous
E[X(m)]{E[r(m)]}-1/2
that
m T co.
tends
results
are obtained
in probability
to zero
< co] > 0. way to estimate
involved
77 in the ergodic
1
2=-
T
T
J X(t) o
could be the best alternative. In M/G/l retrial for E(X] from which we obtain the estimator
fjl
=
case, is via any estimator
of E[X] are not iid, the integral
in the estimation
queues
f of E[X].
estimator
4 we have (see [3]) an explicit
expression
2(1- P)i - Pp2 ’
2xp
expression for the variance of $1 where E[u] = l/X and Pz = E[s2]. W e can get an asymptotic as Rodrigo et al. [6] did for the case when we cannot distinguish between retrials and primary arrivals. REMARK 3.4. In M/G/l observable characteristics
ergodic retrials queues, with exponential retrials, we can use other of the system to estimate n such as the number of calls (primary and/or
repeated) during a service, the length of the busy periods, the number of services occurring in a busy period, the time spent in the system by each customer, etc. The mean values of these characteristics
in steady
state
are known
(see [3,16]) and from them we can obtain
estimators
of n as we did in the previous remark. For instance, if we only observe the number of arrivals during each service and we cannot not distinguish between primary and repeated arrivals, we can estimate the retrial rate p = 71-l using the estimator b=
2(1- P)4- 2P %32
)
where
and Qi, 1 5 i 5 m, is the number
of arrivals
4. CONSTANT
during
the ith service.
FLOW OF REPEATED
CALLS
In the retrial systems studied till now we have assumed that all customers in orbit compete to obtain service, that is, each customer tries his luck again until he finds the service area free and he does this independently of how other customers in orbit behave. We now consider the case when retrials from the orbit occur in an ordered manner resulting in an output flow of repeated calls which is independent of the number of items in the orbit. These retrial systems were introduced by Fayolle [l] and Farahmand (21. We next do an analysis similar to that of Section 2. We assume that conditions (cl) and (~2) hold. Condition (~3) now becomes r-1/2
6
log(1 - H(w’; II)))
JL
0,
T T co a.s.,
(c3)
Large Sample Inference
if T = t is nonrandom.
which is satisfied occurs, {X(t);
this condition
is also satisfied
t 2 0) is nonergodic.
If T = T(m) is the instant
As in Section
For any general
retrial
2, we assume
distribution
-n -5 m
PROOF.
Proceeding
process.
For the nonergodic
that
(cl)
LEMMA 4.2.
max(1, p’),
even when
queue is observed
=mfcQ idle period after esch departure.
as we did in Lemma 2.1 we conclude
Let {X(t);
retrial
and (~3) hold. In this case we have the
that n/m
case, when m is large enough,
5
1 when X(t) is an ergodic
the idle periods
distributed as min(w 1 w > w*,u 1 u > u’) since the orbit Using (2.2), (2.3), and (2.4), the lemma is proved.
distribution
distribution,
a G/G/l
we have that
and I is the stationary
where p* = p + (E[u])-‘E[I]
when the mth departure
at least when H is an Erlangian
over the time interval (O,T(m)] and that conditions asymptotic results given below. LEMMA 4.1.
205
t 2 0) be an ergodic stochastic
is occupied
after departures with
are
probability
one. I
process.
Then,
for any general retrial
we have $ 5
Moreover, when {X(t);
(E[w])-’
Pr[X
2 11,
m t 00.
t > 0) is nonergodic,
$2 Let To and Tl be the amount respectively. We can write
PROOF.
m t 00.
(E[w])-‘,
of time in (0, T] during
which the orbit is empty
and busy,
T _- r/T1 T - T/Tl’ where Tl = w* + WI + .e. + w,. and w* < 00, a.s. Then, r/T1 tends in probability On the other hand, T/T1 tends in probability to Pr[X 2 11, if the system is ergodic,
to E[w]-l. and to one
otherwise.
I
THEOREM 4.1.
under
Let 6, 4, and 6 be the ML estimators
the appropriate
ML estimators
regularity
conditions
6, d, and +j are consistent rnli2(8
obtained
on the density
with asymptotic
- 0) =+ N (0, a_la,2)
,
from likelihood
functions
(1.2).
Then,
f, g, and h (see [141), the
distributions
m1i2($
- 4) +
N (0, c$) ,
and m’i2(ij where a = max(l,p’) x(E[w])-’ PROOF.
and < = E[u](E[w])-1
in the nonergodic
asymptotically
case.
N (~,a:<-‘)
- 77) +
Moreover,
2 l] in the ergodic case, and < = p’E[u] ^ m’i2(0 - 0), m’/2(q4 - 4), and m’i2(7j - r,~)are
Pr[X
independent.
The theorem
and 4.2 instead REMARK 4.1.
,
can be proved proceeding of Lemmas 2.1 and 2.2.
If the densities
as we did in Theorem distributed
-n -%max(l,p+X(X+~)-‘) m pmin(l,Pr[X
with parameters
=a,
2 11) = pb,
m’i2(+j - 7) =S N(o, c), where c = Xp(ab)-‘.
Note that p + X(X + II)-’
4.1 a
f and h are exponentially
& 2
2.1 but using Lemmas
2 1 if X(t)
is nonergodic.
X and ~1,
A. RODRIGO AND M. VP;ZQUEZ
206 REMARK
The case of constant
4.2.
who find the server
retrials
busy wait in a “room”
can be interpreted and the server
in another must
way. The customers
search
for customers
after
a
random idle period which is independent of the number of waiting customers. Note that in this case there are no retrials when the service is busy and therefore it is not exactly the same as the system
just described.
can also consider
priority
If we know the number between
primary
the parameter customers
of customers
and repeated
in orbit
arrivals,
77. For instance, arrivals
are called
“queues
on the number
with priority of customers
at the end of each service
we can easily
if the /ct” idle period
primary
and yk is the duration
and repeated
of q would be the unique
where i = (m--1+x,)/T. estimators
which depend
obtain,
in both
is exp(zcr,/q),
arrivals
solution
of the kth idle period.
and primary
arrivals
can also be used to estimate
the parameter
property
We
(room).
and we can distinguish cases,
an estimator
for of
where r is the number
If we could not distinguish
occur at rate A, the ML estimator
(which exists with probability
Because of the memoryless
intervals”. in orbit
where zk is the number
in orbit at the end of the kth service, we get the ML estimator
of nonprimary between
In [17] these systems intervals
one) of the likelihood
of exponential
7 in the standard
M/G/l
equation
distributions, retrial
these
queue (see
Section 2) with interretrial times exp(l/q). Note that although we need less sample information to get these estimators we cannot use whatever information we might have on unsuccessful retrials.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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