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Operations Research Letters 26 (2000) 231–235
www.elsevier.com/locate/dsw

Some well-behaved estimators for the M= M= 1 queue
Shen Zhenga , Andrew F. Seilab; ∗
a Knowledge
b Department

Engineering Department, Equifax, Inc., Alpharetta, GA 30005, USA
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 =(1
ˆ − )
ˆ for the average
number of customers =(1 − ) in a stationary M=M=1 queueing model is innite (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 innite mean-squared error. Finally, we propose
alternative estimators for these four system performance measures when  ¡ 0 , where 0 ¡ 1 is a known constant, and
show that these alternative estimators are strongly consistent, asymptotically unbiased and have nite variance and nite
c 2000 Elsevier Science B.V. All rights reserved.
mean squared error.
Keywords: M=M=1 queue; Mean waiting time in queue; Mean squared error loss

1. Introduction
Let f(1 ; : : : ; k ) denote the performance measure
for a system model, where 1 ; 2 ; : : : ; k are unknown
parameters of the model. For example, in a stationary M=M=1 queue, a performance measure is the mean
waiting time for customers in the queue, i.e., f(; )=
=( − ), where  is 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 thereCorresponding 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 , respectively. The function f(ˆ1 ; : : : ; ˆk ) is a natural estimator of f(1 ; : : : ; k ). We will call such an estimator
constructed by substituting parameter estimators in a
formula for the system performance measure a substitution estimator.
For the stationary M=M=1 queue, Schruben and
Kulkarni [3] found that, given the observed trac
ˆ ˆ ¡ 1; the expected value of the estiintensity ˆ = =
mator for the mean number of customers in the queue
L = =(1 − ) is innite; that is,

∗

)I[¡1]
] = +∞;
E[E(Lˆ1s |I[¡1]
ˆ
ˆ

c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 3 0 - 4

232

S. Zheng, A.F. Seila / Operations Research Letters 26 (2000) 231–235

where Lˆ1s = =(1
ˆ
− )
ˆ and I[¡1]
is the indicator
ˆ
function of the set [ˆ ¡ 1]. To overcome this problem, 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 substitution estimators for some system performance
measures in the stationary M=M=1 queueing system. We consider the following four system performance 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 alternative estimators for these performance measures
when  ¡ 0 ¡ 1, where 0 is a known constant.

In Section 2, we give the properties for these substitution estimators. In Section 3, the proposed alternative 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 ; : : : ; Xm be an independent and identically distributed (i.i.d.) random
sample of size m from the interarrival time distribution
that is exponential with mean 1=, and let Y1 ; Y2 ; : : : ; Yn
be an i.i.d. random sample of size n from the service time distribution that is exponential with mean
1=. All observations (X1 ; : : : ; Xm ); (Y1 ; : : : ; Yn ) are assumed to be mutually independent.
The sample mean
Pm
interarrival time, X m = i=1 Xi =m, is used to estimate the mean interarrival time 1= and the sample
mean service time, Y n , is used to estimate mean service 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, Wˆ 1q ; Wˆ 1s ; Lˆ1q and Lˆ1s , for
Wq ; Ws ; Lq and Ls , respectively, are:
2

Yn
;
Xm − Yn

(2.1)

X mY n
;
Wˆ 1s =

Xm − Yn

(2.2)

Wˆ 1q =

2

Lˆ1q =

Yn
;
X m (X m − Y n )

(2.3)

Yn
:
(2.4)
(X m − Y n )

The following theorem establishes that, although these
estimators are strongly consistent, their means do not
exist, the expected values of their absolute values are
innite and their mean-squared errors are innite.

Lˆ1s =

Theorem 2.1. For Wˆ 1q ; Wˆ 1s ; Lˆ1q and Lˆ1s dened by
(2:1)–(2:4); for the M=M=1 queue with trac intensity  = = ¡ 1;
a:s:
a:s:
a:s:
Wˆ 1q →; Wq ; Wˆ 1s → Ws ; Lˆ1q → Lq ;
a:s:
Lˆ1s → Ls ;

E Wˆ 1q ;

as m; n → ∞;

E Wˆ 1s ;

E Lˆ1q ;

E Lˆ1s do not exist;

E|Wˆ 1q | = E|Wˆ 1s | = E|Lˆ1q | = E|Lˆ1s | = +∞;

(2.5)
(2.6)
(2.7)

E(Wˆ 1q − Wq )2 = E(Wˆ 1s − Ws )2 = E(Lˆ1q − Lq )2
= E(Lˆ1s − Ls )2 = +∞:

(2.8)

Remark 1. These results hold for  ¡ 1. However,
they can also be shown to hold even if  ¡ 0 ¡ 1:
The proof of Theorem 2.1 is given in Section 4.

One might expect that the reason for the results in
Theorem 2.1 is that it is possible to compute ˆ ¿ 1
from the samples of interarrival and service times.
The following Theorem says that even given [ˆ ¡ 1],
the expected values of the above four estimators on
[ˆ ¡ 1], are all innite.

233

S. Zheng, A.F. Seila / Operations Research Letters 26 (2000) 231–235

Theorem 2.2. For Wˆ 1q ; Wˆ 1s ; Lˆ1q and Lˆ1s dened by
(2:1)–(2:4); for the M=M=1 queue with trac intensity  = = ¡ 1;
)I[¡1]
] = E[E(Wˆ 1s |I[¡1]
)I[¡1]
]
E[E(Wˆ 1q |I[¡1]
ˆ
ˆ

ˆ
ˆ
)I[¡1]
]
= E[E(Lˆ1q |I[¡1]
ˆ
ˆ
)I[¡1]
]
= E[E(Lˆ1s |I[¡1]
ˆ
ˆ
= +∞:
Remark 2. Schruben and Kulkarni [3] showed that
even if the sample space were restricted to samples
for which ˆ ¡ 1, the expected value of Ls is innite,
)I[¡1]
]=+∞. Our proof in Section
i.e., E[E(Lˆs | I[¡1]
ˆ

ˆ
4 is somewhat more straightforward than their’s and
arrives at the same conclusions.

The following theorem establishes the statistical properties of these alternative estimators:
Theorem 3.1. For the estimators dened by (3:1)–
(3:4) in an M=M=1 queue with trac intensity
 ¡ 0 ¡ 1; where 0 is known;
E Wˆ q = Wq + O(1=n) + O(1=m);

(3.5)

E Wˆ s = Ws + O(1=n) + O(1=m);

(3.6)

E Lˆq = Lq + O(1=n) + O(1=m);

(3.7)

E Lˆs = Ls + O(1=n) + O(1=m);

(3.8)

a:s:
a:s:
Wˆ q → Wq ; Wˆ s → Ws ;
a:s:
a:s:
Lˆq → Lq ; Lˆs → Ls ;

as m; n → ∞

(3.9)

and
3. Alternative estimators
The properties of the substitution estimators in Section 2 are a result of the arbitrarily long tails of the
distributions of the estimators. This observation motivated the following alternative estimators for the
above four system performance measures:

2

Yn



if Y n 60 X m
 X −Y
n
m
ˆ
(3.1)
Wq =
 


0

Y n otherwise:

1 − 0

X mY n


if Y n 60 X m


 Xm − Yn
(3.2)
Wˆ s =


Y

n


otherwise:
1 − 0

2

Yn


if Y n 60 X m

 X m (X m − Y n )
ˆ
(3.3)
Lq =


2



0

otherwise:
1 − 0

Yn


if Y n 60 X m

 X −Y
m
n
ˆ
(3.4)
Ls =


0


otherwise:
1 − 0

E(Wˆ q − Wq )2 =

(2 − 2 )2 1
2
1
+ 2
4
( − ) m
 ( − )4 n
+O(1=n2 ) + O(1=m2 ) + O(1=mn)
(3.10)

E(Wˆ s − Ws )2 =

2
2
1
1
+
4
( − ) m ( − )4 n
+O(1=n2 ) + O(1=m2 ) + O(1=mn)
(3.11)

E(Lˆq − Lq )2 =

4 (2 − )2 1
4 (2 − )2 1
+
2 ( − )4 m
2 ( − )4 n
+O(1=n2 ) + O(1=m2 ) + O(1=mn)
(3.12)

E(Lˆs − Ls )2 =

2  2 1
2  2 1
+
( − )4 m ( − )4 n
+O(1=n2 ) + O(1=m2 ) + O(1=mn)
(3.13)

Theorem 3.1 shows that these estimators are
strongly consistent, but that they have nite means
and nite mean-squared error. Moreover, this theorem
establishes the rate of convergence (as m; n → ∞) of
their means and mean-squared errors.

234

S. Zheng, A.F. Seila / Operations Research Letters 26 (2000) 231–235

4. Discussion
While these results show that it is possible to create well-behaved estimators for the four performance
measures, there is an obvious problem in actually using these estimators: What value should be used for
0 ? We make the assumption that  ¡ 0 ¡ 1. However, since  is unknown, an appropriate value for 0
is also unknown.
In practice, the analyst can generally specify an upper bound on the acceptable level of system congestion, i.e., an upper bound on the acceptable values of
. Call this upper bound ∗ . Then, any value of 0 in
the range ∗ ¡ 0 ¡ 1 would allow Theorem 3.1 to be
applied, and a value of 0 close 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 than 0 , then 0 would be
used in place of ˆ in the substitution estimators. If this
event were to occur, however, the assumptions of the
model, i.e., the assumption that the system is stationary, 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
a:s:
a:s:
ˆ → ; ˆ →  and (2.1) – (2.4) are all continuous
functions. To show (2.6), we will only show that E Wˆ q
does not exist; the other results can be proved similarly. Note that X m ∼ (m; m) and Y n ∼ (n; n),
where
(; ) denotes a gamma distribution with parameters  and  and A ∼ 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

K is a positive constant and p(x) is the p.d.f. of the
(m; m) distribution. Observe that
Z 2
Z x n+1
y
+
E Wˆ 1q ¿ K
e−ny dy d x
p(x)
1
1 x−y
Z x
Z 2
1
dy d x
p(x)e−nx
¿K
x
−
y
1
1
Z 2Z x
1
dy d x
¿K
x
−
y
1
1
= +∞
−
Similarly, E Wˆ 1q can be shown to be innite. Hence,
E Wˆ 1q does not exist ([1], p. 84). (2.6), (2.7) and (2.8)
can be proved using the same approach.

Proof of Theorem 2.2. We will prove that E[E(Wˆ 1q |
)I[¡1]
] = +∞. The other results can be proved
I[¡1]
ˆ
ˆ
similarly. Note that by the denition of conditional
expectation (see [1], p. 207),
)I[¡1]
] = E[Wˆ 1q I[¡1]
]
E[E(Wˆ 1q | I[¡1]
ˆ
ˆ
ˆ
+
= E Wˆ 1q

=∞
Proof of Theorem 3.1. We will show (3.10), (3.5)
a:s:
and Wˆ q → Wq . The other conclusions can be proved
similarly.
Observe that
2

Wˆ q =

Yn
0
I
I
+ Yn
;
1 − 0 [Y n ¿0 X m ]
X m − Y n [Y n 60 X m ]

and
2

Yn
a:s:
→ Wq ;
Xm − Yn

as m; n → ∞

a:s:

so for Wˆ q → Wq , it suces to show
a:s:

I[Y n ¿0 X m ] → 0;

as m; n → ∞

(4.1)

E Wˆ 1q = E Wˆ 1q − E Wˆ 1q ;

Let the event Am; n = [Y n ¿ 0 X m ]: Now (4.1) follows
from the fact that

where

lim Am; n = {!: ! ∈ Am; n ; for innite number of m

+

+
E Wˆ 1q = K

Z

∞

Z

∞

−

p(x)

0

−
E Wˆ 1q

=K

0

Z

x

Z

∞

0

p(x)

x

y2 n−1 −ny
dy d x
y e
x−y
y2 n−1 −ny
dy d x;
y e
y−x

and n}
= {!: Y n (!) ¿ 0 X m (!); for innite m
and n}


Yn
¿0 ;
⊂ lim
m; n→∞ X m

S. Zheng, A.F. Seila / Operations Research Letters 26 (2000) 231–235

where ! ∈
and (
; F; P) is an appropriate probability space for {X1 ; X2 ; : : : ; Xn ; Y1 ; Y2 ; : : : ; Yn }, and the
fact that


Yn
¿0 = 0:
Pr lim
m; n→∞ X m
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
60 :
[Y n ∈ D1 (0 ); X m ∈ D2 (0 )] ⊆
Xm
Observe that for any  ¿ 1, and any  ∈ (0; min
(1=; 1=)),
Pr[Y n 6∈ D1 ()] = O(1=n )
and
Pr[X m 6∈ D2 ()] = O(1=m ):

(4.2)

Hence, we have:
0 6 E(Wˆ q − Wq )2 I[Y n 60 X m ]
−E(Wˆ q − Wq )2 I[Y n ∈D1 (0 );X m ∈D2 (0 )]




= O(1=n ) + O(1=m );

235

(4.3), we have, for any  ¿ 1,
E(Wˆ q − Wq )I[Y n 60 X m ]
= E(Wˆ q − Wq )I[Y n ∈D1 (0 ); X m ∈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 Technology and Donald Gross of George Mason University for stimulating and insightful discussions on this
topic. We also wish to thank two anonymous referees for their suggestions. In particular, one of the referees provided useful suggestions regarding how this
methodology could be applied in practice and a second referee provided an improved and more elegant
proof of Theorem 2.1.
References

(4.3)

for any  ¿ 1.
Thus, by expanding for the function Wˆ q in a Taylor
series on [Y n ∈ D1 (0 ); X m ∈ D2 (0 )]; and noting that
the terms involving E(Wˆ q − W q)2 are dominated by
(4.3), we have (3.10). For (3.5), note that similar to

[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.
[3] L. Schruben, R. Kulkarni, Some consequences of estimating
parameters for the M=M=1 queue, Oper. Res. Lett. 1 (1982)
75–78.