Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue1.2000:
Operations Research Letters 27 (2000) 39–46
www.elsevier.com/locate/dsw
On optimal exhaustive policies for the M= G= 1-queue
R.E. Lilloa;∗ , M. Martnb
a Dpto.
de Estadstica y Econometra, Universidad Carlos III de Madrid, Spain
de Estadstica e I.O., Universidad Complutense de Madrid, Spain
b Dpto.
Received 1 April 1998; received in revised form 1 June 2000
Abstract
Given an M=G=1 queue controlled by an exhaustive policy P, we consider a (P + )-policy consisting of turning the server
on at a random time later than P. The objective is to obtain necessary and sucient conditions such that the (P + )-policy
are better than the P-policy. Under the in nite-horizon average-cost criterion, policies are compared when the costs assumed
are linear. When the holding cost is the waiting time cost per unit time per customer, the optimality of the N -policy over,
both the (N + )-policy and the D-policy is showed. We will also discuss on the di erent types of T -policies, single and
c 2000 Elsevier Science
multiple, establishing a relation between them, which is independent of the optimization criterion.
B.V. All rights reserved.
Keywords: Queueing system; Exhaustive policy; N -policy; T -policy; D-policy; Holding cost
1. Introduction
We consider an M=G=1 queueing system with vacation time and exhaustive service discipline. By exhaustive, we mean that customers are continuously served
until there is no customer in the system (see [13]).
By vacation time, we mean that the server becomes
unavailable for occasional intervals of time when the
system becomes empty, (see [14] for a quick review).
Under exhaustive vacation policies, this model has
been studied by Doshi [4], Fuhrmann and Cooper
[5], Kroese and Schmidt [9], Li and Zhu [10] and
Miyazawa [12]. The cost structure considered in these
models classically involves the holding cost, h(s), a
Corresponding author. Fax: +34-91-624-98-49.
E-mail address: lillo@est-econ.uc3m.es (R.E. Lillo).
∗
cost per unit time, which is a function of the state s of
the system; in most vacation models, h(s) is a linear
function of s.
Let P be a general exhaustive policy that controls
when the server should be turned on. We will consider
a modi ed policy of P, consisting in adding a random
vacation time to the initial vacation time associated
to P; being independent of the arrival process. The
modi ed policy is denoted by P+ . Our rst objective
in this paper is to obtain the optimal when P is the
operating policy, i.e., is it possible to choose in such
way that P + is a better policy than P?. A policy
P is better than P′ if the associated average expected
cost per unit time for P is smaller than for P′ .
Assuming that P is a N -policy consisting in turning
the server o when the system becomes empty and
turning it on when N customers (N ¿1) are present in
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 8 - 9
40
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
the system, we obtain that the optimal (N + )-policy is
the optimal N -policy. This type of policies was studied
by Yadin and Naor [15] and Heyman [7].
Balachandran (see [1]) was the rst to introduce a
vacation model in which the control variable of the
system is the workload. The D-policy for controlling
the system as de ned in [1] consists in turning the
server on when the cumulative service times of the
customers in the system exceed the value D and turning it o when the system is empty. In [2,3], it is
proved that the optimal D-policy is better than the
optimal N -policy when the holding cost is the waiting time cost per unit workload per unit time. In this
paper, we compare both policies but considering the
holding cost as the waiting time cost per unit time per
customer. We conclude that the optimal N -policy is at
least as good as the optimal D-policy. Finally, a discussion regarding di erent types of T -policies is included in the last section. We will show the optimality
of the single T -policy over the multiple T -policy for
every optimization criterion.
2. The control problem
Consider a single-server queueing system in which
customers arrive according to a Poisson process with
parameter . The service times of customers are i.i.d.
random variables having a common general distribution function S(t) with nite rst and second moments, s1 and s2 . Assume that the service discipline
is non-preemptive and the service order is FIFO. Let
= s1 , and to ensure stability, assume that ¡ 1.
The economics of system operation is in
uenced
by various costs involved: (1) Running costs, r1 (r0 ),
a cost per unit time when the server is on (o ). (2)
Switching cost, R1 (R2 ), a non-negative set-up (shut
down) cost, incurred each time the server is turned on
(o ). R = R1 + R2 denotes the total switching cost. (3)
Serving cost. Gain s, per customer served regardless
of the service time. (4) Holding cost, a penalty, h,
per customer in the system per unit time. To avoid
triviality, it is assumed h ¿ 0.
The running costs can be negative since idle periods
or vacation time could be used for additional tasks.
Hence, these models have wide applicability in analyzing many computer systems, data communication
networks and production systems.
Every possible policy of turning the server on and
o during the operation horizon for the queueing system leads to a di erent operating cost. We need a criterion to compare these policies through a cost function.
In this paper, we consider the average total expected
cost per unit time, which is used in many practical
models such as inventory models (see [6]). De ne a
busy cycle as the time between successive vacation
times. Since the busy cycles form a renewal process,
the total cost rate is
R + r0 E[T0 ] + r1 E[T −T0 ] + sE[V ] + hE[W ]
c(P) =
;
E[T ]
(1)
where T0 is the time during a busy cycle in which the
server is o , and T is the duration of a busy cycle.
T − T0 is commonly referred to as the occupation period, that is, the total time elapsing from the moment
the server returns from a vacation until it departs for
another one. V is the number of served customers during T , and W denotes the total holding time (accumulated waiting time) during a busy cycle. All these
variables depend on the operating policy P.
For a xed policy P and a busy cycle, let NP be
the number of customers present at the opening, and
let TP be the duration of the buildup period before the
opening of the channel. In [11], it is proved that an
exhaustive policy for the M=G=1 queue satis es the
property,
E[NP ] = E[TP ]:
(2)
In the following, let n(P) = E[NP ].
3. Improvement of an exhaustive policy
We consider now a variation of an exhaustive
P-policy. In this variation the vacation time that the
server spends when the system becomes empty is
TP + , where is a random variable with nite rst
and second moments 1 and 2 . Moreover, is independent of both, TP and the arrival process. We refer
to this exhaustive policy as the (P + )-policy. In this
section, we analyze when the (P + )-policy is better
than the initial P-policy in relation to the total cost
per unit time given in (1).
We rstly need some notation. Let Tn ; n¿1 be a
occupation period that begins with n customers present
41
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
in the system, and let Vn ; n¿1 be the number of customers served during Tn . Let n (n ) be the expected
value of Tn (Vn ). It is known that
Theorem 3.1. If P is an exhaustive policy; c(P) can
be rewritten as
n = n; n = n;
c(P) = K +
for all n¿1;
where
Let Wn ; n¿1 be the accumulated waiting time during
Tn , and let !n = E[Wn ].
Lemma 3.1. Assume a positive recurrent M=G=1-queue.
Then; the sequence {!n ; n¿0} is nite if and only
if s2 ¡ ∞. In this case;
s2
;
2(1 − )2
!n = n!1 +
n(n − 1)
2
(3)
K = r0 (1 − ) + r1 + s + h
2 s2
:
2(1 − )
Proof. Considering that P satis es (2), and the
stream is a Poisson process, it is easy to see that
nP
; E[T − T0 ] = nP ; E[V ] = nP : (7)
E[WP ] can be obtained by conditioning arguments
E[T0 ] =
NP (NP − 1)
+ !NP ;
2
where the rst term of the right-hand side is the accumulated waiting time during the buildup period before
the opening of the channel. Unconditioning NP and
considering (4), we get
E[WP | NP ] =
for n ¿ 1:
(4)
Proof. The case n = 1 can be obtained by conditioning on the number of customers to arrive during the
occupation period. Then, !1 = , where denotes
the expected waiting time of a customer in the system
before entering service, given that he arrives during a
occupation period. For the M=G=1 queue,
=
(6)
where
1
=
:
1−
s1
;
=
1−
!1 =
R(1 − )
V [NP ] + n2P − nP
+h
;
nP
2nP
s2
:
2(1 − )
V (NP ) + n2P − nP
2
(1 − ):
(8)
Substituting (7) and (8) into (1) and simplifying, we
get c(P) such as in (6).
Hence, the value of !1 is given by (3). If n ¿ 1, choose
a customer among the n present in the system and
consider a new queue with only the xed customer at
time t = 0. Then, the following equation is obvious:
∗
;
Wn = W1 + (n − 1)T1 + Wn−1
E[WP ] = !1 nP +
(5)
d
∗
∗
is independent of
= Wn−1 , and Wn−1
where Wn−1
(W1 ; T1 ). From (5), it easy to see
!n = !1 + (n − 1) + !n−1
and by successive substitutions,
n(n − 1)
;
2
which completes the proof.
The purpose of this section is to nd conditions such
that c(P + )6c(P). Note that the costs r0 ; r1 and
s are irrelevant in (6) when the optimization problem
consists of comparing di erent policies through the
same objective function.
Theorem 3.2. The policy P + improves the policy
P if and only if
2n2P + nP ¡ E[NP2 ] +
!n = n!1 +
In the following result, we obtain an expression for
(1) that shows explicitly the in
uence of the exhaustive policy P on the objective function.
2R(1 − )
:
h
(9)
Proof. Note that NP+ = NP + N and that both variables are independent. From (6), we need to calculate
E[NP+ ] = nP + 1 ;
V [NP+ ] = V [NP ] + V [N ]:
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R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
Proof. To optimize c(P + 1 ) as a function of 1 ¿ 0;
we have to solve the equation dc(P + 1 )=d 1 = 0;
which is equivalent to solving
Conditioning by ; we have
V [N ] = E[V [N | ]] + V [E[N | ]]
h
2 h 2
+ hnP 1 + hn2P − (V [NP ] + n2P − nP )
2 1
2
−R(1 − ) = 0;
= E[ ] + V [ ]
= 1 + 2 2 − 2 21 :
Thus,
and hence we obtain
p
−nP ± V [NP ] − nP + 2R(1 − )=h
:
1 =
V [NP+ ] + (E[NP+ ])2 − E[NP+ ]
=V [NP ] + n2P − nP + 2 2 + 2 1 nP :
Since 2 ¿ 21 and this value appears in the numerator
of (6), the optimal choice of is to take 2 = 21 ; which
implies that is a constant. In this case, c(P + )
becomes
c(P + ) = K +
+h
R(1 − )
nP + 1
4. The N -policy
V [NP ] + n2P − nP + 2 21 + 2 1 nP
:
2(nP + 1 )
(10)
Comparing c(P) with c(P + ) and after routine algebra, we have
c(P) − c(P + ) ¿ 0 ⇔ 1 nP ¡ V [NP ]
2R(1 − )h
:
−n2P − nP +
h
Since 1 ¿ 0; P + improves P if and only if we can
choose a positive value of 1 that satis es the second
inequality. This is always possible if,
V [NP ] − n2P − nP +
Since P satis es condition (9), the discriminant is
non-negative. Di erentiating twice, we prove that the
minimum is achieved if the positive root is considered,
and therefore a value of opt
1 ¿ 0 exists.
2R(1 − )
¿ 0;
h
and replacing V [NP ] by E[NP2 ] − n2P ; we obtain condition (9).
Now, we are interested in obtaining the optimal
value of 1 when the condition of improvement is ful lled.
Corollary 3.1. If a P-policy satis es (9); the optimal
opt
P + -policy is P + opt
1 ; 1 being determined by
p
−nP + V [NP ] − nP + 2R(1 − )=h
opt
: (11)
1 =
Much of the research in queueing theory has been
concerned with design and control problems, and
speci cally, with optimization. Yadin and Naor [15]
were the rst to introduce a queueing system with a
removable server applying a N -policy. The N -policy
is to turn the server on when the queue size reaches
the number N , and to turn the server o when the
system is empty. Heyman [7] also considered similar
policies and showed the optimality of the above policy under certain conditions. In this section, we prove
the optimality of the N -policy over the (N + )-policy.
First, we rewrite (6) as a function of N :
c(N ) = K +
(h=2)(N 2 − N ) + R(1 − )
:
N
Let N opt be the optimal N -policy. For the average
criterion, Heyman [7] proved that the optimal value
of N is one of the two integers surrounding the value,
r
2R(1 − )
′
:
(12)
N =
h
Therefore, N opt is either [N ′ ] or [N ′ ] + 1; where [x]
is the largest integer contained in x. By comparing
c([N ′ ]) with c([N ′ ] + 1); we have
[N ′ ]
′
′
′
[N
]
if
N
−
[N
]¿
;
[N ′ ] + N ′
opt
(13)
N =
′
[N ′ ] + 1 if N ′ − [N ′ ] ¡ [N ] :
[N ′ ] + N ′
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
Hence, c(N ) can be rewritten as
N ′2
h
N −1+
:
c(N ) = K +
2
N
(14)
The arguments of the last section lead to the next
result.
Theorem 4.1. The N -policy can be improved by a
(N + )-policy if and only if
p
−1 + 1 + 4(2R(1 − )=h)
:
(15)
0¡N ¡
2
Proof. It is sucient to see when condition (9) is
ful lled. Then, considering that nP = N and E[NP2 ] =
N 2 ; (9) is the same as
2R(1 − )
¡ 0:
(16)
h
Since N ¿0; we only consider the positive root of
the quadratic equation. Thus, the N -policy can be improved if and only if
p
−1 + 1 + 4(2R(1 − )=h)
;
0¡N ¡
2
and the proof is complete.
N2 + N −
Theorem 4.2. The optimal N -policy; N opt cannot be
improved by any (N opt + )-policy.
Proof. It is sucient to evaluate (16) for N = N opt .
Note that (16) can be rewritten as
N 2 + N − N ′2 ¡ 0;
(17)
where N ′ is given in (12). Now, we substitute the
possible values for N opt . If N opt = [N ′ ] + 1; we have
that
(N opt )2 + N opt − N ′2 = ([N ′ ] + 1 − N ′ )
× ([N ′ ] + 1 + N ′ ) + [N ′ ] + 1;
and it is easy to see that this expression is positive,
which implies that (17) does not hold. If N opt = [N ′ ];
we have that
(N opt )2 +N opt −N ′2 = ([N ′ ]−N ′ )([N ′ ]+N ′ )+[N ′ ];
and this expression is positive due to (13). Then, (17)
is not satis ed. This implies that the N opt -policy is
better than the N opt + -policy.
43
We now want to seek the best (N + )-policy, optimizing jointly on N and . Let c(N; 1 ) be the cost
rate depending on N and . From (10), we have
h N 2 − N + 2 21 + 2 1 N
c(N; 1 ) = K +
2
N + 1
+
R(1 − )
:
N + 1
(18)
Theorem 4.3. The optimal (N + )-policy is the
N opt -policy.
Proof. Di erentiating in (18) with respect to 1 and
equating to zero, we obtain an equation for the optimal
1 as function of N; i.e.,
√
−N + N ′2 − N
opt
:
(19)
1 (N ) =
By straightforward calculations, 1 ¿ 0 if and only if
(16) holds, that is, if the N -policy can be improved.
This implies that if N ¿N ′ the optimal (N + )-policy
is the N -policy. Thus, we focus on the values of N that
satisfy (15). For such N; the optimal cost is determined
by the pair (N; opt
1 (N )); as follows:
c(N; opt (N )) = 2(N ′2 − N )1=2 :
(20)
Note that (20) is a decreasing function of N . This implies that we only have to consider the largest possible value of N satisfying N ¡ N opt . There are two
cases. If N opt = [N ′ ]; we have to compare c([N ′ ] − 1;
′
opt
). By routine algebra, we get
opt
1 ([N ]−1)) and c(N
′
opt
c([N ′ ] − 1; opt
)⇔
1 ([N ] − 1))6c(N
1 ¡ [(N ′2 + 1 − [N ′ ])1=2 − [N ′ ]]2 :
Hence, using (13), it is easy to see that the inequality
does not hold. Now, if N opt = [N ′ ] + 1; we have to
′
opt
compare c([N ′ ]; opt
). After calcu1 ([N ])) and c(N
lations, we have
′
opt
)⇔1
c([N ′ ]; opt
1 ([N ]))6c(N
¡ [(N ′2 − [N ′ ])1=2 − ([N ′ ] + 1)]2
and using (13) again, we can see that the inequality
does not hold. Thus, the optimal (N + )-policy is the
N opt -policy.
44
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
5. The D-policy
We consider another type of policies introduced in
the literature on vacations models. Balachandran [1]
was the rst to introduce the D-policy consisting in
turning the server on when the workload reaches the
value D and turning it o when the system is empty. If
h is now a holding cost per unit time per unit workload,
Balachandran and Tijms [2] obtained that the optimal
value of D is determined by the equation
Z D∗
R(1 − )
∗
;
E[Mx ] d x =
D +
h
0
where Mx denote the number of customers present at
the opening of the station if the accumulated workload
is x. Boxma [3] generalizes the results given in [2], and
proves that the D-policy is better than the N -policy. In
this section, we analyze the D-policy considering the
holding cost as in the previous sections, that is, a cost
per unit time per customer in the system. Moreover,
we compare the D-policy and the N -policy.
Let S(·)
P∞be the distribution of service times and
m(y) = n=1 S n (y); the renewal function, where S n
is the n-fold convolution of S with itself. Observe that
E[MD ] = 1 + m(D). The cost rate associated with this
type of policies is function on D¿0. Then, (6) can be
rewritten as
R(1 − )
h E[MD2 ] − E[MD ]
:
+
c(D) = K +
2
E[MD ]
E[MD ]
(21)
Theorem 5.1. The optimal N -policy is better than
the optimal D-policy.
Proof. Let the service durations of customers be a
constant, S. In this case, E[MD ] = 1 + [D=S] and (21)
becomes
N ′2
h
[D=S] +
:
(22)
c(D) = K +
2
[D=S] + 1
Comparing (22) and (14), we nd that c(D) = c(N ),
when N =[D=S]+1. Hence, the policies are equivalent
in the following sense: if we take E[MD ] = N opt , we
obtain the minimum value of c(D) that we denote by
Dopt . Then, the optimal policies for the D and the N
strategies lead to the same cost rate when the service
times are constant, i.e., c(N opt ) = c(Dopt ). Now, if the
service times are not constant, we get
h
N ′2
c(D)¿K +
E[MD ] − 1 +
;
2
E[MD ]
(23)
and again, the right-hand side of the inequality is
minimum when E[MD ] = N opt , which proves the superiority of the N -policy over the D-policy in the
M=G=1-system.
We now proceed to nd an expression for E[MD2 ]
and then substitute it in (21). To nd E[MD2 ], conditioning on the duration of the rst service S1 = u, we
have
Z D
2
2
E[MD−u
] dS(u):
E[MD ] = (1 + 2m(D)) +
0
Solving this renewal equation, we get that
E[MD2 ] = 1 + 3m(D) + 2m ∗ m(D);
where m ∗ m means convolution. Then, substituting in
(21) and simplifying, the average expected cost per
unit time for the D-policy yields
N ′2
h 2m(D) + 2m ∗ m(D)
:
+
c(D) = K +
2
1 + m(D)
1 + m(D)
(24)
If m(D) is di erentiable, we have that Dopt is the solution to the equation
m′ (D) + (1 + m(D))(m ∗ m′ (D))
m′ (D)N ′2
= 0:
(25)
−m′ (D)(m ∗ m(D)) −
2
For the special case in which the service durations
of customers are exponentially distributed with rate ,
we can give an explicit expression of Dopt . We have
that
E[MD ] = 1 + D;
m ∗ m(D) =
2 D2
;
2
m ∗ m′ (D) = 2 D:
Then, we have to solve
2 D2 + 2D + 2 − N ′2 = 0:
Hence,
D
opt
=
−1 +
√
N ′2 − 1
:
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
45
Note that this value di ers from the value of D obtained in [2] with h de ned as the cost per unit workload per unit time.
let T be an r.v. with distribution function F. Setting
as
(
T − X if X 6T;
=
0
if X ¿ T;
6. The T-policy
P is shown to be an (1 + )-policy. Thus, we have
(ST) ⊂ (1 + ). Similarly, we prove (1 + ) ⊂ (ST).
Then, (1 + ) = (ST) and the proof is complete.
There is abundant literature discussing queueing
systems controlled by T -policies, for instance, by
Fuhrmann and Cooper, [5], Heyman [8], Doshi [4]
and Li and Zhu [10]. In this mode of operation, whenever the system becomes empty, the server starts
a vacation (a random time T ), independent of the
arrival process. If the queue is still empty upon its
return, two alternatives are commonly considered:
(a) The single vacation-model, ST-policy. A busy
period begins at the time of the rst arrival.
(b) The multiple vacation-model, MT-policy. The
server takes another equally distributed vacation
and continue in this manner until he nds at least
one waiting customer upon return from a vacation.
Heyman [8] shows that under the asymptotic average criterion, the optimal N -policy is always better than the optimal MT-policy. In this section we
complete this information by comparing ST, MT, and
(1 + )-policies. We mean by (1 + )-policy, the (N +
)-policy with N = 1. Let (ST), (MT), and (1 + ),
be the collection of all ST-policies, MT-policies, and
(1 + )-policies, respectively. The following relations
among these sets of strategies can be derived.
Theorem 6.1. (a) (MT) ⊂(1+ ); but (1+ ) ⊂
6 (MT).
(b) (ST) = (1 + ).
Proof. Let P be a MT-policy that is characterized
by the distribution function F(·) of a single random
vacation. Let Tn ; n=1; 2; : : : be i.i.d. random variables
with distribution function F. Let X be the waiting time
for the rst customer after the server was turned o .
Recall that X is independent of the sequence Tn ; n¿1.
Hence, P can be considered as an (1 + )-policy with
de ned as follows: if T1 + · · · + Tn−1 ¡ X 6T1 +
· · · + Tn ; n¿1, set = T1 + · · · + Tn − X . This proves
that (MT) ⊂ (1 + ). On the other hand, it is obvious
that (1 + ) ⊂
6 (MT). Now, let P be an ST-policy, and
Remark 6.1. As an immediate consequence of Theorem 6.1, we have obtained that for every cost function and every optimization criterion (1) optimal
ST-policies and (1 + )-policies lead to the same
value of the minimum cost, and (2) the ST-policy
and the (1 + )-policy are better than the MT-policy.
7. Concluding remarks
In this paper we have shown how the cost rate associated to M=G=1 queueing systems controlled by exhaustive policies can be improved under certain conditions by adding an extra constant vacation time.
We proved the optimality of the N -policy over the
(N + )-policy, and over the D-policy. We obtained
the equation that determines the optimal value of D,
and nally, we established relations among the single T -policy, the multiple T -policy, and the family of
(1 + )-policies. These relations hold for every criterion and for every cost structure.
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system, Oper. Res. 16 (1968) 362–382.
[8] D. Heyman, The T -policy for the M/G/1 queue, Mgmt. Sci.
23 (7) (1977) 775–778.
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R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
[9] D.P. Kroese, V. Schmidt, A continuous polling system with
general service times, Ann. Appl. Probab. 2 (1992) 906–927.
[10] H. Li, Y. Zhu, Analysis of M/G/1 queues with delayed
vacations and exhaustive service discipline, European J. Oper.
Res. 92 (1996) 125–134.
[11] R.E. Lillo, Optimal operating policy for an M/G/1 exhaustive
server-vacation model, Methodol. Comput. Appl. Probab.
(2000), forthcoming.
[12] M. Miyazawa, Descomposition formulas for single server
queues with vacations: a uni ed approach by the rate
conservation law, Stochastic Models 10 (1994) 389–413.
[13] H. Takagi, Queueing Analysis, A Foundation of Performance
Evaluation, Vol. 1: Vacation and Priority Systems, Elsevier,
Amsterdam, 1991.
[14] J. Teghem, Control of the service process in a queueing
system, European J. Oper. Res. 23 (1986) 141–158.
[15] M. Yadin, P. Naor, Queueing systems with a removable
service station, Oper. Res. Quart. 14 (1963) 393–405.
www.elsevier.com/locate/dsw
On optimal exhaustive policies for the M= G= 1-queue
R.E. Lilloa;∗ , M. Martnb
a Dpto.
de Estadstica y Econometra, Universidad Carlos III de Madrid, Spain
de Estadstica e I.O., Universidad Complutense de Madrid, Spain
b Dpto.
Received 1 April 1998; received in revised form 1 June 2000
Abstract
Given an M=G=1 queue controlled by an exhaustive policy P, we consider a (P + )-policy consisting of turning the server
on at a random time later than P. The objective is to obtain necessary and sucient conditions such that the (P + )-policy
are better than the P-policy. Under the in nite-horizon average-cost criterion, policies are compared when the costs assumed
are linear. When the holding cost is the waiting time cost per unit time per customer, the optimality of the N -policy over,
both the (N + )-policy and the D-policy is showed. We will also discuss on the di erent types of T -policies, single and
c 2000 Elsevier Science
multiple, establishing a relation between them, which is independent of the optimization criterion.
B.V. All rights reserved.
Keywords: Queueing system; Exhaustive policy; N -policy; T -policy; D-policy; Holding cost
1. Introduction
We consider an M=G=1 queueing system with vacation time and exhaustive service discipline. By exhaustive, we mean that customers are continuously served
until there is no customer in the system (see [13]).
By vacation time, we mean that the server becomes
unavailable for occasional intervals of time when the
system becomes empty, (see [14] for a quick review).
Under exhaustive vacation policies, this model has
been studied by Doshi [4], Fuhrmann and Cooper
[5], Kroese and Schmidt [9], Li and Zhu [10] and
Miyazawa [12]. The cost structure considered in these
models classically involves the holding cost, h(s), a
Corresponding author. Fax: +34-91-624-98-49.
E-mail address: lillo@est-econ.uc3m.es (R.E. Lillo).
∗
cost per unit time, which is a function of the state s of
the system; in most vacation models, h(s) is a linear
function of s.
Let P be a general exhaustive policy that controls
when the server should be turned on. We will consider
a modi ed policy of P, consisting in adding a random
vacation time to the initial vacation time associated
to P; being independent of the arrival process. The
modi ed policy is denoted by P+ . Our rst objective
in this paper is to obtain the optimal when P is the
operating policy, i.e., is it possible to choose in such
way that P + is a better policy than P?. A policy
P is better than P′ if the associated average expected
cost per unit time for P is smaller than for P′ .
Assuming that P is a N -policy consisting in turning
the server o when the system becomes empty and
turning it on when N customers (N ¿1) are present in
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 8 - 9
40
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
the system, we obtain that the optimal (N + )-policy is
the optimal N -policy. This type of policies was studied
by Yadin and Naor [15] and Heyman [7].
Balachandran (see [1]) was the rst to introduce a
vacation model in which the control variable of the
system is the workload. The D-policy for controlling
the system as de ned in [1] consists in turning the
server on when the cumulative service times of the
customers in the system exceed the value D and turning it o when the system is empty. In [2,3], it is
proved that the optimal D-policy is better than the
optimal N -policy when the holding cost is the waiting time cost per unit workload per unit time. In this
paper, we compare both policies but considering the
holding cost as the waiting time cost per unit time per
customer. We conclude that the optimal N -policy is at
least as good as the optimal D-policy. Finally, a discussion regarding di erent types of T -policies is included in the last section. We will show the optimality
of the single T -policy over the multiple T -policy for
every optimization criterion.
2. The control problem
Consider a single-server queueing system in which
customers arrive according to a Poisson process with
parameter . The service times of customers are i.i.d.
random variables having a common general distribution function S(t) with nite rst and second moments, s1 and s2 . Assume that the service discipline
is non-preemptive and the service order is FIFO. Let
= s1 , and to ensure stability, assume that ¡ 1.
The economics of system operation is in
uenced
by various costs involved: (1) Running costs, r1 (r0 ),
a cost per unit time when the server is on (o ). (2)
Switching cost, R1 (R2 ), a non-negative set-up (shut
down) cost, incurred each time the server is turned on
(o ). R = R1 + R2 denotes the total switching cost. (3)
Serving cost. Gain s, per customer served regardless
of the service time. (4) Holding cost, a penalty, h,
per customer in the system per unit time. To avoid
triviality, it is assumed h ¿ 0.
The running costs can be negative since idle periods
or vacation time could be used for additional tasks.
Hence, these models have wide applicability in analyzing many computer systems, data communication
networks and production systems.
Every possible policy of turning the server on and
o during the operation horizon for the queueing system leads to a di erent operating cost. We need a criterion to compare these policies through a cost function.
In this paper, we consider the average total expected
cost per unit time, which is used in many practical
models such as inventory models (see [6]). De ne a
busy cycle as the time between successive vacation
times. Since the busy cycles form a renewal process,
the total cost rate is
R + r0 E[T0 ] + r1 E[T −T0 ] + sE[V ] + hE[W ]
c(P) =
;
E[T ]
(1)
where T0 is the time during a busy cycle in which the
server is o , and T is the duration of a busy cycle.
T − T0 is commonly referred to as the occupation period, that is, the total time elapsing from the moment
the server returns from a vacation until it departs for
another one. V is the number of served customers during T , and W denotes the total holding time (accumulated waiting time) during a busy cycle. All these
variables depend on the operating policy P.
For a xed policy P and a busy cycle, let NP be
the number of customers present at the opening, and
let TP be the duration of the buildup period before the
opening of the channel. In [11], it is proved that an
exhaustive policy for the M=G=1 queue satis es the
property,
E[NP ] = E[TP ]:
(2)
In the following, let n(P) = E[NP ].
3. Improvement of an exhaustive policy
We consider now a variation of an exhaustive
P-policy. In this variation the vacation time that the
server spends when the system becomes empty is
TP + , where is a random variable with nite rst
and second moments 1 and 2 . Moreover, is independent of both, TP and the arrival process. We refer
to this exhaustive policy as the (P + )-policy. In this
section, we analyze when the (P + )-policy is better
than the initial P-policy in relation to the total cost
per unit time given in (1).
We rstly need some notation. Let Tn ; n¿1 be a
occupation period that begins with n customers present
41
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
in the system, and let Vn ; n¿1 be the number of customers served during Tn . Let n (n ) be the expected
value of Tn (Vn ). It is known that
Theorem 3.1. If P is an exhaustive policy; c(P) can
be rewritten as
n = n; n = n;
c(P) = K +
for all n¿1;
where
Let Wn ; n¿1 be the accumulated waiting time during
Tn , and let !n = E[Wn ].
Lemma 3.1. Assume a positive recurrent M=G=1-queue.
Then; the sequence {!n ; n¿0} is nite if and only
if s2 ¡ ∞. In this case;
s2
;
2(1 − )2
!n = n!1 +
n(n − 1)
2
(3)
K = r0 (1 − ) + r1 + s + h
2 s2
:
2(1 − )
Proof. Considering that P satis es (2), and the
stream is a Poisson process, it is easy to see that
nP
; E[T − T0 ] = nP ; E[V ] = nP : (7)
E[WP ] can be obtained by conditioning arguments
E[T0 ] =
NP (NP − 1)
+ !NP ;
2
where the rst term of the right-hand side is the accumulated waiting time during the buildup period before
the opening of the channel. Unconditioning NP and
considering (4), we get
E[WP | NP ] =
for n ¿ 1:
(4)
Proof. The case n = 1 can be obtained by conditioning on the number of customers to arrive during the
occupation period. Then, !1 = , where denotes
the expected waiting time of a customer in the system
before entering service, given that he arrives during a
occupation period. For the M=G=1 queue,
=
(6)
where
1
=
:
1−
s1
;
=
1−
!1 =
R(1 − )
V [NP ] + n2P − nP
+h
;
nP
2nP
s2
:
2(1 − )
V (NP ) + n2P − nP
2
(1 − ):
(8)
Substituting (7) and (8) into (1) and simplifying, we
get c(P) such as in (6).
Hence, the value of !1 is given by (3). If n ¿ 1, choose
a customer among the n present in the system and
consider a new queue with only the xed customer at
time t = 0. Then, the following equation is obvious:
∗
;
Wn = W1 + (n − 1)T1 + Wn−1
E[WP ] = !1 nP +
(5)
d
∗
∗
is independent of
= Wn−1 , and Wn−1
where Wn−1
(W1 ; T1 ). From (5), it easy to see
!n = !1 + (n − 1) + !n−1
and by successive substitutions,
n(n − 1)
;
2
which completes the proof.
The purpose of this section is to nd conditions such
that c(P + )6c(P). Note that the costs r0 ; r1 and
s are irrelevant in (6) when the optimization problem
consists of comparing di erent policies through the
same objective function.
Theorem 3.2. The policy P + improves the policy
P if and only if
2n2P + nP ¡ E[NP2 ] +
!n = n!1 +
In the following result, we obtain an expression for
(1) that shows explicitly the in
uence of the exhaustive policy P on the objective function.
2R(1 − )
:
h
(9)
Proof. Note that NP+ = NP + N and that both variables are independent. From (6), we need to calculate
E[NP+ ] = nP + 1 ;
V [NP+ ] = V [NP ] + V [N ]:
42
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
Proof. To optimize c(P + 1 ) as a function of 1 ¿ 0;
we have to solve the equation dc(P + 1 )=d 1 = 0;
which is equivalent to solving
Conditioning by ; we have
V [N ] = E[V [N | ]] + V [E[N | ]]
h
2 h 2
+ hnP 1 + hn2P − (V [NP ] + n2P − nP )
2 1
2
−R(1 − ) = 0;
= E[ ] + V [ ]
= 1 + 2 2 − 2 21 :
Thus,
and hence we obtain
p
−nP ± V [NP ] − nP + 2R(1 − )=h
:
1 =
V [NP+ ] + (E[NP+ ])2 − E[NP+ ]
=V [NP ] + n2P − nP + 2 2 + 2 1 nP :
Since 2 ¿ 21 and this value appears in the numerator
of (6), the optimal choice of is to take 2 = 21 ; which
implies that is a constant. In this case, c(P + )
becomes
c(P + ) = K +
+h
R(1 − )
nP + 1
4. The N -policy
V [NP ] + n2P − nP + 2 21 + 2 1 nP
:
2(nP + 1 )
(10)
Comparing c(P) with c(P + ) and after routine algebra, we have
c(P) − c(P + ) ¿ 0 ⇔ 1 nP ¡ V [NP ]
2R(1 − )h
:
−n2P − nP +
h
Since 1 ¿ 0; P + improves P if and only if we can
choose a positive value of 1 that satis es the second
inequality. This is always possible if,
V [NP ] − n2P − nP +
Since P satis es condition (9), the discriminant is
non-negative. Di erentiating twice, we prove that the
minimum is achieved if the positive root is considered,
and therefore a value of opt
1 ¿ 0 exists.
2R(1 − )
¿ 0;
h
and replacing V [NP ] by E[NP2 ] − n2P ; we obtain condition (9).
Now, we are interested in obtaining the optimal
value of 1 when the condition of improvement is ful lled.
Corollary 3.1. If a P-policy satis es (9); the optimal
opt
P + -policy is P + opt
1 ; 1 being determined by
p
−nP + V [NP ] − nP + 2R(1 − )=h
opt
: (11)
1 =
Much of the research in queueing theory has been
concerned with design and control problems, and
speci cally, with optimization. Yadin and Naor [15]
were the rst to introduce a queueing system with a
removable server applying a N -policy. The N -policy
is to turn the server on when the queue size reaches
the number N , and to turn the server o when the
system is empty. Heyman [7] also considered similar
policies and showed the optimality of the above policy under certain conditions. In this section, we prove
the optimality of the N -policy over the (N + )-policy.
First, we rewrite (6) as a function of N :
c(N ) = K +
(h=2)(N 2 − N ) + R(1 − )
:
N
Let N opt be the optimal N -policy. For the average
criterion, Heyman [7] proved that the optimal value
of N is one of the two integers surrounding the value,
r
2R(1 − )
′
:
(12)
N =
h
Therefore, N opt is either [N ′ ] or [N ′ ] + 1; where [x]
is the largest integer contained in x. By comparing
c([N ′ ]) with c([N ′ ] + 1); we have
[N ′ ]
′
′
′
[N
]
if
N
−
[N
]¿
;
[N ′ ] + N ′
opt
(13)
N =
′
[N ′ ] + 1 if N ′ − [N ′ ] ¡ [N ] :
[N ′ ] + N ′
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
Hence, c(N ) can be rewritten as
N ′2
h
N −1+
:
c(N ) = K +
2
N
(14)
The arguments of the last section lead to the next
result.
Theorem 4.1. The N -policy can be improved by a
(N + )-policy if and only if
p
−1 + 1 + 4(2R(1 − )=h)
:
(15)
0¡N ¡
2
Proof. It is sucient to see when condition (9) is
ful lled. Then, considering that nP = N and E[NP2 ] =
N 2 ; (9) is the same as
2R(1 − )
¡ 0:
(16)
h
Since N ¿0; we only consider the positive root of
the quadratic equation. Thus, the N -policy can be improved if and only if
p
−1 + 1 + 4(2R(1 − )=h)
;
0¡N ¡
2
and the proof is complete.
N2 + N −
Theorem 4.2. The optimal N -policy; N opt cannot be
improved by any (N opt + )-policy.
Proof. It is sucient to evaluate (16) for N = N opt .
Note that (16) can be rewritten as
N 2 + N − N ′2 ¡ 0;
(17)
where N ′ is given in (12). Now, we substitute the
possible values for N opt . If N opt = [N ′ ] + 1; we have
that
(N opt )2 + N opt − N ′2 = ([N ′ ] + 1 − N ′ )
× ([N ′ ] + 1 + N ′ ) + [N ′ ] + 1;
and it is easy to see that this expression is positive,
which implies that (17) does not hold. If N opt = [N ′ ];
we have that
(N opt )2 +N opt −N ′2 = ([N ′ ]−N ′ )([N ′ ]+N ′ )+[N ′ ];
and this expression is positive due to (13). Then, (17)
is not satis ed. This implies that the N opt -policy is
better than the N opt + -policy.
43
We now want to seek the best (N + )-policy, optimizing jointly on N and . Let c(N; 1 ) be the cost
rate depending on N and . From (10), we have
h N 2 − N + 2 21 + 2 1 N
c(N; 1 ) = K +
2
N + 1
+
R(1 − )
:
N + 1
(18)
Theorem 4.3. The optimal (N + )-policy is the
N opt -policy.
Proof. Di erentiating in (18) with respect to 1 and
equating to zero, we obtain an equation for the optimal
1 as function of N; i.e.,
√
−N + N ′2 − N
opt
:
(19)
1 (N ) =
By straightforward calculations, 1 ¿ 0 if and only if
(16) holds, that is, if the N -policy can be improved.
This implies that if N ¿N ′ the optimal (N + )-policy
is the N -policy. Thus, we focus on the values of N that
satisfy (15). For such N; the optimal cost is determined
by the pair (N; opt
1 (N )); as follows:
c(N; opt (N )) = 2(N ′2 − N )1=2 :
(20)
Note that (20) is a decreasing function of N . This implies that we only have to consider the largest possible value of N satisfying N ¡ N opt . There are two
cases. If N opt = [N ′ ]; we have to compare c([N ′ ] − 1;
′
opt
). By routine algebra, we get
opt
1 ([N ]−1)) and c(N
′
opt
c([N ′ ] − 1; opt
)⇔
1 ([N ] − 1))6c(N
1 ¡ [(N ′2 + 1 − [N ′ ])1=2 − [N ′ ]]2 :
Hence, using (13), it is easy to see that the inequality
does not hold. Now, if N opt = [N ′ ] + 1; we have to
′
opt
compare c([N ′ ]; opt
). After calcu1 ([N ])) and c(N
lations, we have
′
opt
)⇔1
c([N ′ ]; opt
1 ([N ]))6c(N
¡ [(N ′2 − [N ′ ])1=2 − ([N ′ ] + 1)]2
and using (13) again, we can see that the inequality
does not hold. Thus, the optimal (N + )-policy is the
N opt -policy.
44
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
5. The D-policy
We consider another type of policies introduced in
the literature on vacations models. Balachandran [1]
was the rst to introduce the D-policy consisting in
turning the server on when the workload reaches the
value D and turning it o when the system is empty. If
h is now a holding cost per unit time per unit workload,
Balachandran and Tijms [2] obtained that the optimal
value of D is determined by the equation
Z D∗
R(1 − )
∗
;
E[Mx ] d x =
D +
h
0
where Mx denote the number of customers present at
the opening of the station if the accumulated workload
is x. Boxma [3] generalizes the results given in [2], and
proves that the D-policy is better than the N -policy. In
this section, we analyze the D-policy considering the
holding cost as in the previous sections, that is, a cost
per unit time per customer in the system. Moreover,
we compare the D-policy and the N -policy.
Let S(·)
P∞be the distribution of service times and
m(y) = n=1 S n (y); the renewal function, where S n
is the n-fold convolution of S with itself. Observe that
E[MD ] = 1 + m(D). The cost rate associated with this
type of policies is function on D¿0. Then, (6) can be
rewritten as
R(1 − )
h E[MD2 ] − E[MD ]
:
+
c(D) = K +
2
E[MD ]
E[MD ]
(21)
Theorem 5.1. The optimal N -policy is better than
the optimal D-policy.
Proof. Let the service durations of customers be a
constant, S. In this case, E[MD ] = 1 + [D=S] and (21)
becomes
N ′2
h
[D=S] +
:
(22)
c(D) = K +
2
[D=S] + 1
Comparing (22) and (14), we nd that c(D) = c(N ),
when N =[D=S]+1. Hence, the policies are equivalent
in the following sense: if we take E[MD ] = N opt , we
obtain the minimum value of c(D) that we denote by
Dopt . Then, the optimal policies for the D and the N
strategies lead to the same cost rate when the service
times are constant, i.e., c(N opt ) = c(Dopt ). Now, if the
service times are not constant, we get
h
N ′2
c(D)¿K +
E[MD ] − 1 +
;
2
E[MD ]
(23)
and again, the right-hand side of the inequality is
minimum when E[MD ] = N opt , which proves the superiority of the N -policy over the D-policy in the
M=G=1-system.
We now proceed to nd an expression for E[MD2 ]
and then substitute it in (21). To nd E[MD2 ], conditioning on the duration of the rst service S1 = u, we
have
Z D
2
2
E[MD−u
] dS(u):
E[MD ] = (1 + 2m(D)) +
0
Solving this renewal equation, we get that
E[MD2 ] = 1 + 3m(D) + 2m ∗ m(D);
where m ∗ m means convolution. Then, substituting in
(21) and simplifying, the average expected cost per
unit time for the D-policy yields
N ′2
h 2m(D) + 2m ∗ m(D)
:
+
c(D) = K +
2
1 + m(D)
1 + m(D)
(24)
If m(D) is di erentiable, we have that Dopt is the solution to the equation
m′ (D) + (1 + m(D))(m ∗ m′ (D))
m′ (D)N ′2
= 0:
(25)
−m′ (D)(m ∗ m(D)) −
2
For the special case in which the service durations
of customers are exponentially distributed with rate ,
we can give an explicit expression of Dopt . We have
that
E[MD ] = 1 + D;
m ∗ m(D) =
2 D2
;
2
m ∗ m′ (D) = 2 D:
Then, we have to solve
2 D2 + 2D + 2 − N ′2 = 0:
Hence,
D
opt
=
−1 +
√
N ′2 − 1
:
R.E. Lillo, M. Martn / Operations Research Letters 27 (2000) 39–46
45
Note that this value di ers from the value of D obtained in [2] with h de ned as the cost per unit workload per unit time.
let T be an r.v. with distribution function F. Setting
as
(
T − X if X 6T;
=
0
if X ¿ T;
6. The T-policy
P is shown to be an (1 + )-policy. Thus, we have
(ST) ⊂ (1 + ). Similarly, we prove (1 + ) ⊂ (ST).
Then, (1 + ) = (ST) and the proof is complete.
There is abundant literature discussing queueing
systems controlled by T -policies, for instance, by
Fuhrmann and Cooper, [5], Heyman [8], Doshi [4]
and Li and Zhu [10]. In this mode of operation, whenever the system becomes empty, the server starts
a vacation (a random time T ), independent of the
arrival process. If the queue is still empty upon its
return, two alternatives are commonly considered:
(a) The single vacation-model, ST-policy. A busy
period begins at the time of the rst arrival.
(b) The multiple vacation-model, MT-policy. The
server takes another equally distributed vacation
and continue in this manner until he nds at least
one waiting customer upon return from a vacation.
Heyman [8] shows that under the asymptotic average criterion, the optimal N -policy is always better than the optimal MT-policy. In this section we
complete this information by comparing ST, MT, and
(1 + )-policies. We mean by (1 + )-policy, the (N +
)-policy with N = 1. Let (ST), (MT), and (1 + ),
be the collection of all ST-policies, MT-policies, and
(1 + )-policies, respectively. The following relations
among these sets of strategies can be derived.
Theorem 6.1. (a) (MT) ⊂(1+ ); but (1+ ) ⊂
6 (MT).
(b) (ST) = (1 + ).
Proof. Let P be a MT-policy that is characterized
by the distribution function F(·) of a single random
vacation. Let Tn ; n=1; 2; : : : be i.i.d. random variables
with distribution function F. Let X be the waiting time
for the rst customer after the server was turned o .
Recall that X is independent of the sequence Tn ; n¿1.
Hence, P can be considered as an (1 + )-policy with
de ned as follows: if T1 + · · · + Tn−1 ¡ X 6T1 +
· · · + Tn ; n¿1, set = T1 + · · · + Tn − X . This proves
that (MT) ⊂ (1 + ). On the other hand, it is obvious
that (1 + ) ⊂
6 (MT). Now, let P be an ST-policy, and
Remark 6.1. As an immediate consequence of Theorem 6.1, we have obtained that for every cost function and every optimization criterion (1) optimal
ST-policies and (1 + )-policies lead to the same
value of the minimum cost, and (2) the ST-policy
and the (1 + )-policy are better than the MT-policy.
7. Concluding remarks
In this paper we have shown how the cost rate associated to M=G=1 queueing systems controlled by exhaustive policies can be improved under certain conditions by adding an extra constant vacation time.
We proved the optimality of the N -policy over the
(N + )-policy, and over the D-policy. We obtained
the equation that determines the optimal value of D,
and nally, we established relations among the single T -policy, the multiple T -policy, and the family of
(1 + )-policies. These relations hold for every criterion and for every cost structure.
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