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Some well-behaved estimators for the M
=
M
=
1 queue
Shen Zheng
a, Andrew F. Seila
b;∗aKnowledge Engineering Department, Equifax, Inc., Alpharetta, GA 30005, USA
bDepartment of MIS, Terry College of Business, University of Georgia, Athens, GA 30602-6273, USA Received 1 June 1996; received in revised form 1 February 2000
Abstract
It is known that, given the observed trac intensity ˆ ¡1, the expected value of the estimator ˆ=(1−) for the averageˆ number of customers=(1−) in a stationary M=M=1 queueing model is innite (Schruben and Kulkarni, Oper. Res. Lett. 1 (1982) 75 –78). In this paper we generalize the above ndings to other system performance measures. Second, we show that, for the following four system performance measures: (a) mean waiting time in queue, (b) mean waiting time in system, (c) mean number of customers in queue and (d) mean number of customers in the system, estimators constructed by substituting parameter estimators for unknown parameters in the formula for the performance measure all have the undesirable properties that the expected value of the estimator does not exist and the estimator has innite mean-squared error. Finally, we propose alternative estimators for these four system performance measures when ¡ 0, where0¡1 is a known constant, and show that these alternative estimators are strongly consistent, asymptotically unbiased and have nite variance and nite mean squared error. c2000 Elsevier Science B.V. All rights reserved.
Keywords:M=M=1 queue; Mean waiting time in queue; Mean squared error loss
1. Introduction
Letf(1; : : : ; k) denote the performance measure
for a system model, where1; 2; : : : ; kare unknown
parameters of the model. For example, in a station-ary M=M=1 queue, a performance measure is the mean waiting time for customers in the queue, i.e.,f(; )=
=(−), whereis the arrival rate,is the service
rate and== ¡1 is the condition for stationarity. In order to use the model, the unknown values of the parameters 1; 2; : : : ; k must be known, and
there-∗Corresponding author. +1-706 583-0037.
E-mail addresses: sz96@yahoo.com (S. Zheng), aseila@ terry.uga.edu (A.F. Seila).
fore must be estimated from observed system data. Let ˆ1; : : : ;ˆk be estimators of 1; 2; : : : ; k,
respec-tively. The functionf( ˆ1; : : : ;ˆk) is a natural estima-tor of f(1; : : : ; k). We will call such an estimator
constructed by substituting parameter estimators in a formula for the system performance measure a substi-tution estimator.
For the stationary M=M=1 queue, Schruben and Kulkarni [3] found that, given the observed trac intensity ˆ= ˆ= ¡ˆ 1;the expected value of the esti-mator for the mean number of customers in the queue
L==(1−) is innite; that is,
E[E( ˆL1s|I[ ˆ¡1])I[ ˆ¡1]] = +∞;
where ˆL1s = ˆ=(1 −) andˆ I[ ˆ¡1] is the indicator
function of the set [ ˆ ¡1]. To overcome this prob-lem, they suggested using an estimator that uses an estimate of trac intensity that is strictly less than 1. In this paper we extend the work of Schruben and Kulkarni by presenting some properties of these sub-stitution estimators for some system performance measures in the stationary M=M=1 queueing sys-tem. We consider the following four system perfor-mance measures: (a) the mean waiting time in queue, (b) the mean waiting time in the system, (c) the mean number of customers in the queue and (d) the mean number of customers in the system. We propose al-ternative estimators for these performance measures when ¡ 0¡1, where0 is a known constant.
In Section 2, we give the properties for these substi-tution estimators. In Section 3, the proposed alterna-tive estimators are presented and their properties are discussed. The proofs for the results in Sections 2 and 3 are given in Section 4.
2. Substitution estimators for system performance measures for the M=M=1 queue
For the M=M=1 queue, let X1; X2; : : : ; Xmbe an
in-dependent and identically distributed (i.i.d.) random sample of sizemfrom the interarrival time distribution that is exponential with mean 1=, and letY1; Y2; : : : ; Yn
be an i.i.d. random sample of size n from the ser-vice time distribution that is exponential with mean 1=. All observations (X1; : : : ; Xm);(Y1; : : : ; Yn) are
as-sumed to be mutually independent. The sample mean interarrival time, Xm=Pmi=1 Xi=m, is used to
esti-mate the mean interarrival time 1= and the sample mean service time,Yn, is used to estimate mean
ser-vice time 1=. For this stationary queueing system, we also assume that the trac intensity== ¡ 0¡1,
where 0 is a known constant. Expressions for the
steady-state parameters of the M=M=1 queue can be found in Gross and Harris [2]. The mean waiting time in queue is
Wq==(−);
the mean number of customers in queue is
Lq=2=(−);
the mean waiting time in system is
Ws= 1=(−);
and the mean number of customers in the system is
Ls==(−):
The substitution estimators, ˆW1q;Wˆ1s;Lˆ1qand ˆL1s, for
Wq; Ws; Lq andLs, respectively, are:
ˆ
W1q=
Y2n Xm−Yn
; (2.1)
ˆ
W1s=
XmYn
Xm−Yn
; (2.2)
ˆ L1q=
Y2n Xm(Xm−Yn)
; (2.3)
ˆ L1s=
Yn
(Xm−Yn)
: (2.4)
The following theorem establishes that, although these estimators are strongly consistent, their means do not exist, the expected values of their absolute values are innite and their mean-squared errors are innite.
Theorem 2.1. For Wˆ1q;Wˆ1s;Lˆ1q and Lˆ1s dened by
(2:1)–(2:4);for theM=M=1 queue with trac
inten-sity== ¡1;
ˆ W1q
a:s:
→; Wq; Wˆ1s a:s:
→Ws; Lˆ1q a:s: →Lq;
ˆ L1s
a:s:
→Ls; as m; n→ ∞; (2.5)
EWˆ1q; EWˆ1s; ELˆ1q; ELˆ1s do not exist; (2.6)
E|Wˆ1q|=E|Wˆ1s|=E|Lˆ1q|=E|Lˆ1s|= +∞; (2.7)
E( ˆW1q−Wq)2=E( ˆW1s−Ws)2=E( ˆL1q−Lq)2
=E( ˆL1s−Ls)2= +∞: (2.8)
Remark 1. These results hold for ¡1. However, they can also be shown to hold even if ¡ 0¡1:
Theorem 2.2. For Wˆ1q;Wˆ1s;Lˆ1q and Lˆ1s dened by
(2:1)–(2:4); for theM=M=1queue with trac
inten-sity== ¡1;
E[E( ˆW1q|I[ ˆ¡1])I[ ˆ¡1]] =E[E( ˆW1s|I[ ˆ¡1])I[ ˆ¡1]]
=E[E( ˆL1q|I[ ˆ¡1])I[ ˆ¡1]]
=E[E( ˆL1s|I[ ˆ¡1])I[ ˆ¡1]]
= +∞:
Remark 2. Schruben and Kulkarni [3] showed that even if the sample space were restricted to samples for which ˆ ¡1, the expected value ofLs is innite,
i.e.,E[E( ˆLs|I[ ˆ¡1])I[ ˆ¡1]]=+∞. Our proof in Section
4 is somewhat more straightforward than their’s and arrives at the same conclusions.
3. Alternative estimators
The properties of the substitution estimators in Sec-tion 2 are a result of the arbitrarily long tails of the distributions of the estimators. This observation mo-tivated the following alternative estimators for the above four system performance measures:
ˆ
The following theorem establishes the statistical prop-erties of these alternative estimators:
Theorem 3.1. For the estimators dened by (3:1)–
(3:4) in an M=M=1 queue with trac intensity
¡ 0¡1;where0 is known;
4. Discussion
While these results show that it is possible to cre-ate well-behaved estimators for the four performance measures, there is an obvious problem in actually us-ing these estimators: What value should be used for 0? We make the assumption that ¡ 0¡1.
How-ever, sinceis unknown, an appropriate value for0
is also unknown.
In practice, the analyst can generally specify an up-per bound on the acceptable level of system conges-tion, i.e., an upper bound on the acceptable values of
. Call this upper bound∗. Then, any value of
0in
the range∗¡
0¡1 would allow Theorem 3.1 to be
applied, and a value of0close to 1 would reduce the
bias in the estimators. If, after collecting samples of interarrival times and service times, the sample traf-c intensity, ˆ, is larger than0, then 0 would be
used in place of ˆin the substitution estimators. If this event were to occur, however, the assumptions of the model, i.e., the assumption that the system is station-ary, would be called into question and the data would be discarded and resampled, or even the model itself would need to be reexamined.
5. Proofs of Theorems in Section 2 and 3
Proof of Theorem 2.1. (2.5) is evident since ˆ
→a:s: ; ˆ→a:s: and (2.1) – (2.4) are all continuous functions. To show (2.6), we will only show thatEWˆq
does not exist; the other results can be proved simi-larly. Note thatXm∼ (m; m) andYn ∼ (n; n),
where(; ) denotes a gamma distribution with
pa-rameters and andA∼B means that the random
variable A has the same distribution as the random variable B. Since (X1; : : : ; Xm) and (Y1; : : : ; Yn) are
independent, we can write
EWˆ1q=EWˆ
K is a positive constant and p(x) is the p.d.f. of the (m; m) distribution. Observe that
EWˆ+1q¿ K
can be proved using the same approach.
Proof of Theorem 2.2. We will prove thatE[E( ˆW1q|
I[ ˆ¡1])I[ ˆ¡1]] = +∞. The other results can be proved
similarly. Note that by the denition of conditional expectation (see [1], p. 207),
E[E( ˆW1q|I[ ˆ¡1])I[ ˆ¡1]] =E[ ˆW1qI[ ˆ¡1]]
=EWˆ+1q
=∞
Proof of Theorem 3.1. We will show (3.10), (3.5) and ˆWq
a:s:
→Wq. The other conclusions can be proved
similarly.
from the fact that
where!∈ and (; F; P) is an appropriate proba-bility space for{X1; X2; : : : ; Xn; Y1; Y2; : : : ; Yn}, and the
fact that
Pr
lim
m; n→∞
Yn
Xm
¿0
= 0:
For (3.10), let D1(0) = [1= −0;1= +0], and
D2(0) = [1=−0;1=+0].
Note that there exists an 0 ∈ (0;min(1=;1=));
such that
[Yn ∈D1(0); Xm∈D2(0)]⊆
Yn
Xm
60
:
Observe that for any ¿1, and any ∈(0;min (1=;1=)),
Pr[Yn6∈D1()] = O(1=n)
and
Pr[Xm6∈D2()] = O(1=m): (4.2)
Hence, we have:
06E( ˆWq−Wq)2I[Yn60Xm]
−E( ˆWq−Wq)2I[Yn∈D1(0);Xm∈D2(0)]
= O(1=n) + O(1=m); (4.3)
for any ¿1.
Thus, by expanding for the function ˆWqin a Taylor
series on [Yn∈D1(0); Xm∈D2(0)];and noting that
the terms involvingE( ˆWq−Wq)2 are dominated by
(4.3), we have (3.10). For (3.5), note that similar to
(4.3), we have, for any ¿1,
E( ˆWq−Wq)I[Yn60Xm]
=E( ˆWq−Wq)I[Yn∈D1(0); Xm∈D2(0)]
+ O(1=n) + O(1=m)
= O(1=n) + O(1=m): (4.4)
(3.5) follows from (4.2) and (4.4).
Acknowledgements
The authors wish to thank David Goldsman and Christos Alexopoulos of Georgia Institute of Tech-nology and Donald Gross of George Mason Univer-sity for stimulating and insightful discussions on this topic. We also wish to thank two anonymous refer-ees for their suggestions. In particular, one of the ref-erees provided useful suggestions regarding how this methodology could be applied in practice and a sec-ond referee provided an improved and more elegant proof of Theorem 2.1.
References
[1] Y.S. Chow, H. Teicher, Probability Theory, 2nd Edition, Springer, New York, 1988.
[2] D. Gross, C.M. Harris, Fundamentals of Queueing Theory, 2nd Edition, Wiley, New York, 1997.
