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On optimal exhaustive policies for the M
=
G
=
1-queue
R.E. Lillo
a;∗, M. Martn
baDpto. de Estadstica y Econometra, Universidad Carlos III de Madrid, Spain bDpto. de Estadstica e I.O., Universidad Complutense de Madrid, Spain
Received 1 April 1998; received in revised form 1 June 2000
Abstract
Given an M=G=1 queue controlled by an exhaustive policyP, we consider a (P+)-policy consisting of turning the server on at a random timelater thanP. The objective is to obtain necessary and sucient conditions such that the (P+)-policy are better than theP-policy. Under the innite-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 theN-policy over, both the (N+)-policy and theD-policy is showed. We will also discuss on the dierent types ofT-policies, single and multiple, establishing a relation between them, which is independent of the optimization criterion. c2000 Elsevier Science 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 vaca-tion time and exhaustive service discipline. By exhaus-tive, 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 statesof the system; in most vacation models,h(s) is a linear function ofs.
LetP be a general exhaustive policy that controls when the server should be turned on. We will consider a modied policy ofP, consisting in adding a random vacation timeto the initial vacation time associated toP; being independent of the arrival process. The modied policy is denoted byP+. Our rst objective in this paper is to obtain the optimalwhenPis the operating policy, i.e., is it possible to choosein such way thatP+is a better policy thanP?. A policy Pis better thanP′
if the associated average expected cost per unit time forPis smaller than forP′
. Assuming thatPis aN-policy consisting in turning the server o when the system becomes empty and turning it on whenNcustomers (N¿1) are present in
the system, we obtain that the optimal (N+)-policy is the optimalN-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. TheD-policy for controlling the system as dened in [1] consists in turning the server on when the cumulative service times of the customers in the system exceed the valueDand turn-ing it o when the system is empty. In [2,3], it is proved that the optimal D-policy is better than the optimalN-policy when theholding costis the wait-ing 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 optimalN-policy is at least as good as the optimalD-policy. Finally, a dis-cussion regarding dierent types of T-policies is in-cluded in the last section. We will show the optimality of thesingleT-policy over themultipleT-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 distribu-tion funcdistribu-tion S(t) with nite rst and second mo-ments, s1 ands2. 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 inuenced 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+R2denotes the total switching cost. (3) Serving cost. Gains, 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 assumedh ¿0.
Therunning costscan be negative since idle periods or vacation time could be used for additional tasks. Hence, these models have wide applicability in ana-lyzing 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 sys-tem leads to a dierent operating cost. We need a crite-rion 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]). Dene a busy cycle as the time between successive vacation times. Since the busy cycles form a renewal process, the total cost rate is
c(P) =R+r0E[T0] +r1E[T−T0] +sE[V] +hE[W] E[T] ;
(1)
whereT0is the time during a busy cycle in which the
server is o, and T is the duration of a busy cycle. T−T0is commonly referred to as the occupation
pe-riod, that is, the total time elapsing from the moment the server returns from a vacation until it departs for another one.Vis the number of served customers dur-ing T, and W denotes the total holding time (accu-mulated waiting time) during a busy cycle. All these variables depend on the operating policyP.
For a xed policyP and a busy cycle, letNP be
the number of customers present at the opening, and letTPbe 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 satises the property,
E[NP] =E[TP]: (2)
In the following, letn(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+, whereis a random variable with nite rst
and second moments1 and2. Moreover,is
inde-pendent of both,TPand 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. LetTn; n¿1 be a
in the system, and letVn; n¿1 be the number of
cus-tomers served duringTn. Letn (n) be the expected
value ofTn (Vn). It is known that
n=n; n=n; for alln¿1;
where
= s1
1−; = 1 1−:
LetWn; n¿1 be the accumulated waiting time during
Tn, and let!n=E[Wn].
Lemma 3.1. Assume a positive recurrentM=G=1-queue. Then; the sequence{!n; n¿0} is nite if and only
ifs2¡∞.In this case;
!1=
s2
2(1−)2; (3)
!n=n!1+
n(n−1)
2 forn ¿1: (4)
Proof. The casen= 1 can be obtained by condition-ing on the number of customers to arrive durcondition-ing 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,
= s2 2(1−):
Hence, the value of!1is given by (3). Ifn ¿1, choose
a customer among the n present in the system and consider a new queue with only the xed customer at timet= 0. Then, the following equation is obvious: Wn=W1+ (n−1)T1+Wn∗−1; (5)
where W∗
n−1 d
=Wn−1, and Wn∗−1 is independent of
(W1; T1). From (5), it easy to see
!n=!1+ (n−1)+!n−1 and by successive substitutions,
!n=n!1+
n(n−1) 2 ; which completes the proof.
In the following result, we obtain an expression for (1) that shows explicitly the inuence of the exhaus-tive policyPon the objective function.
Theorem 3.1. IfPis an exhaustive policy; c(P)can be rewritten as
c(P) =K+R(1−) nP
+hV[NP] +n
2
P−nP
2nP
; (6)
where
K=r0(1−) +r1+s+h
2s 2
2(1−):
Proof. Considering that P satises (2), and the stream is a Poisson process, it is easy to see that
E[T0] =
nP
; E[T −T0] =nP; E[V] =nP: (7) E[WP] can be obtained by conditioning arguments
E[WP|NP] =
NP(NP−1)
2 +!NP;
where the rst term of the right-hand side is the accu-mulated waiting time during the buildup period before the opening of the channel. Unconditioning NP and
considering (4), we get
E[WP] =!1nP+
V(NP) +n2P−nP
2
(1−):
(8)
Substituting (7) and (8) into (1) and simplifying, we getc(P) such as in (6).
The purpose of this section is to nd conditions such that c(P+)6c(P). Note that the costs r0; r1 and sare irrelevant in (6) when the optimization problem consists of comparing dierent policies through the same objective function.
Theorem 3.2. The policyP+improves the policy Pif and only if
2n2P+nP¡ E[NP2] +
2R(1−)
h : (9)
Proof. Note thatNP+=NP+Nand that both
varia-bles are independent. From (6), we need to calculate
E[NP+] =nP+1;
Conditioning by;we have
choose a positive value of1 that satises the second
inequality. This is always possible if,
V[NP]−n2P−nP+
2R(1−) h ¿0;
and replacingV[NP] byE[NP2]−n2P;we obtain
con-dition (9).
Now, we are interested in obtaining the optimal value of1when the condition of improvement is
ful-lled.
Corollary 3.1. If aP-policy satises(9);the optimal P+-policy isP+opt
1 ; opt
1 being determined by
opt1 =−nP+ which is equivalent to solving
2h and hence we obtain
1=−
nP±
p
V[NP]−nP+ 2R(1−)=h
:
Since P satises condition (9), the discriminant is non-negative. Dierentiating twice, we prove that the minimum is achieved if the positive root is considered, and therefore a value ofopt1 ¿0 exists.
4. TheN-policy
Much of the research in queueing theory has been concerned with design and control problems, and specically, with optimization. Yadin and Naor [15] were the rst to introduce a queueing system with a removable server applying aN-policy. TheN-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 poli-cy under certain conditions. In this section, we prove the optimality of theN-policy over the (N+)-policy. First, we rewrite (6) as a function ofN:
c(N) =K+(h=2)(N
2−N) +R(1−)
N :
LetNopt be the optimalN-policy. For the average
criterion, Heyman [7] proved that the optimal value ofN is one of the two integers surrounding the value,
N′ =
r
2R(1−)
h : (12) Therefore,Nopt is either [N′
] or [N′
Hence,c(N) can be rewritten as
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
0¡ N ¡−1 +
p
1 + 4(2R(1−)=h)
2 : (15)
Proof. It is sucient to see when condition (9) is fullled. Then, considering thatnP=N andE[NP2] =
N2;(9) is the same as
N2+N−2R(1−)
h ¡0: (16) Since N¿0; we only consider the positive root of the quadratic equation. Thus, theN-policy can be im-proved if and only if
0¡ N ¡−1 +
p
1 + 4(2R(1−)=h)
2 ;
and the proof is complete.
Theorem 4.2. The optimalN-policy; Nopt cannot be improved by any(Nopt+)-policy.
Proof. It is sucient to evaluate (16) forN=Nopt.
Note that (16) can be rewritten as
N2+N−N′2
¡0; (17)
where N′
is given in (12). Now, we substitute the possible values forNopt. IfNopt= [N′
and it is easy to see that this expression is positive, which implies that (17) does not hold. IfNopt= [N′
and this expression is positive due to (13). Then, (17) is not satised. This implies that the Nopt-policy is
better than theNopt+-policy.
We now want to seek the best (N+)-policy, op-timizing jointly onN and. Letc(N; 1) be the cost
rate depending onN and. From (10), we have
c(N; 1) =K+
Theorem 4.3. The optimal (N + )-policy is the Nopt-policy.
Proof. Dierentiating in (18) with respect to1 and
equating to zero, we obtain an equation for the optimal 1 as function ofN;i.e.,
opt1 (N) =−N+
√
N′2−N
: (19) By straightforward calculations,1¿0 if and only if
(16) holds, that is, if theN-policy can be improved. This implies that ifN¿N′
the optimal (N+)-policy is theN-policy. Thus, we focus on the values ofNthat satisfy (15). For suchN;the optimal cost is determined by the pair (N; opt1 (N));as follows:
c(N; opt(N)) = 2(N′2
−N)1=2: (20)
Note that (20) is a decreasing function ofN. This im-plies that we only have to consider the largest pos-sible value ofN satisfyingN ¡ Nopt. There are two
cases. IfNopt= [N′
Hence, using (13), it is easy to see that the inequality does not hold. Now, ifNopt= [N′
5. TheD-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 valueDand turning it o when the system is empty. If his now a holding cost per unit time per unit workload, Balachandran and Tijms [2] obtained that the optimal value ofDis determined by the equation
D∗
whereMxdenote the number of customers present at
the opening of the station if the accumulated workload isx. Boxma [3] generalizes the results given in [2], and proves that theD-policy is better than theN-policy. In this section, we analyze theD-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 theD-policy and theN-policy.
Let S(·) be the distribution of service times and m(y) =P∞
n=1Sn(y); the renewal function, whereSn
is then-fold convolution ofSwith itself. Observe that E[MD] = 1 +m(D). The cost rate associated with this
type of policies is function onD¿0. Then, (6) can be rewritten as
Theorem 5.1. The optimal N-policy is better than the optimalD-policy.
Proof. Let the service durations of customers be a constant,S. In this case,E[MD] = 1 + [D=S] and (21) whenN=[D=S]+1. Hence, the policies are equivalent in the following sense: if we takeE[MD] =Nopt, we
obtain the minimum value ofc(D) that we denote by Dopt. Then, the optimal policies for theDand theN
strategies lead to the same cost rate when the service times are constant, i.e.,c(Nopt) =c(Dopt). Now, if the
service times are not constant, we get
c(D)¿K+h
and again, the right-hand side of the inequality is minimum whenE[MD] =Nopt, which proves the
su-periority of the N-policy over the D-policy in the M=G=1-system.
We now proceed to nd an expression forE[M2
D]
and then substitute it in (21). To ndE[M2
D],
condi-tioning on the duration of the rst serviceS1=u, we
have
E[MD2] = (1 + 2m(D)) +
Z D
0
E[MD2−u] dS(u):
Solving this renewal equation, we get that
E[MD2] = 1 + 3m(D) + 2m∗m(D);
wherem∗mmeans convolution. Then, substituting in (21) and simplifying, the average expected cost per unit time for theD-policy yields
c(D) =K+h
Ifm(D) is dierentiable, we have thatDopt is the
so-lution to the equation
m′ For the special case in which the service durations of customers are exponentially distributed with rate, we can give an explicit expression ofDopt. We have
that Then, we have to solve
Note that this value diers from the value of D ob-tained in [2] withhdened as the cost per unit work-load per unit time.
6. TheT-policy
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, when-ever 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 vaca-tion.
Heyman [8] shows that under the asymptotic ave-rage criterion, the optimal N-policy is always bet-ter 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 withN = 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. LetTn; n=1;2; : : :be i.i.d. random variables
with distribution functionF. LetX be the waiting time for the rst customer after the server was turned o. Recall thatX is independent of the sequenceTn; n¿1.
Hence,Pcan be considered as an (1 +)-policy with dened as follows: ifT1+· · ·+Tn−1¡ X6T1+
· · ·+Tn; n¿1, set=T1+· · ·+Tn−X. This proves
that (MT)⊂(1 +). On the other hand, it is obvious that (1 +)6⊂(MT). Now, letPbe an ST-policy, and
letT be an r.v. with distribution functionF. Setting as
=
(
T −X if X6T; 0 if X ¿ T;
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.
Remark 6.1. As an immediate consequence of The-orem 6.1, we have obtained that for every cost func-tion and every optimizafunc-tion 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 asso-ciated to M=G=1 queueing systems controlled by ex-haustive policies can be improved under certain con-ditions 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 ofD, and nally, we established relations among the sin-gleT-policy, the multipleT-policy, and the family of (1 +)-policies. These relations hold for every crite-rion and for every cost structure.
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