Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol28.Issue1.2001:

Operations Research Letters 28 (2001) 35–49
www.elsevier.com/locate/dsw

An analysis of multiple-class vacation queues with individual
thresholds
Ho Woo Leea; ∗ , Won Joo Seoa , Seung Hyun Yoonb
a Department

of Systems Management Engineering, Sung Kyun Kwan University, Su Won, South Korea
b ETRI, Dae Jon, South Korea
Received 15 July 1998; received in revised form 1 August 2000

Abstract
This study examines a multiple-class M=G=1 queue with multiple thresholds. Each class of customer arrives according
to an independent Poisson process, and each class has its own threshold. The idle server is reactivated as soon as any one
of the thresholds is reached. Both FCFS and non-preemptive priority cases are considered. The Laplace–Stieltjes transform
c 2001
of the waiting time distribution function and the mean waiting time for each class of customers are derived.
Elsevier Science B.V. All rights reserved.
Keywords: Queues; Multiple class; Priority; N -policy

1. Introduction
Most studies on N -policy use a single threshold. This has been the tendency for studies involving multiple
class queues. In these models, the idle server is reactivated when the total number of customers (including all
classes) reaches a threshold (for an example of this, see [12]). In this paper a queueing system is proposed in
which each class has its own threshold and the idle server is reactivated as soon as any one of the thresholds
is reached. This paper was motivated by a real production system in which a machine needed to process
multiple types of products.
The rst study on N -policy was carried out by Yadin and Naor [14]. They obtained the mean waiting time
and the mean queue length, and then derived the optimal N ∗ that minimizes the overall average operating
cost. Hofri [4] studied two N -policy queues attended by a single server. Lee et al. [9] considered a threshold
policy in which the server is reactivated when the number of the start-up class reached its threshold, regardless
Corresponding author.
E-mail address: hwlee@yurim.skku.ac.kr (H.W. Lee).

∗

c 2001 Elsevier Science B.V. All rights reserved.
0167-6377/01/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 6 4 - X

36

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

of the number of customers in other classes. For more studies on N -policy and related works, readers are
advised to see Takagi [12].
The queueing system related to N -policy can be thought of as a vacation system. Vacation queues have
attracted much attention from numerous researchers since Levy and Yechiali [10] and Heyman [3]. Kella
[5] studied the M=G=1 queue with N -policy and vacations. Batch arrival queues under N -policy, with and
without multiple vacations, were rst studied by Lee and Srinivasan [8]. Lee et al. [6,7] considered batch
arrival queues with N -policy and vacations, and derived the queue length distribution in decomposed forms.
The decomposition property was rst proved by Fuhrmann and Cooper [2] and then by Shanthikumar [11].
For a comprehensive survey on vacation queues, readers are advised to see Doshi [1] and Takagi [12].

2. The system and notation
A system with the following specications is considered in this study:
(1) There are r classes.
(2) Customers of class p; (p = 1; 2; : : : ; r) arrive according to a Poisson process with rate p independently
of other classes.
(3) Service times within a class are identically and independently distributed. Dierent classes have dierent

service time distributions, but they are independent of each other.
(4) There is a single server and the queue capacity is unlimited.
(5) If no priorities are applied, FCFS is assumed regardless of the classes. If priorities are applied, nonpreemption is assumed. Class-i has higher priority over class-j, for i ¡ j. FCFS is applied within a class.
(6) Class-p has a threshold of Np .
(7) The idle server is reactivated as soon as any one of the thresholds is reached.
Let us dene the following notations. For the purpose of convenience, the test customer is assumed to belong
to class-p. In most cases, the notations used by Takagi [12] are adopted, but some minor modications are
made.
HC
EHC
EC
LC
BPIP
∗
()
Wq(p)
E(Wq(p) )
p P
r
= P

k=1 k
p
p+ = Pk=1 k
r
p− = k=p+1 k
Sp
Bp (x)
Bp∗ ()
B∗ ()

high-priority customer, i.e., a customer who belongs to one of the classes {1; 2; : : : ; p − 1}
equal- or high-priority customer, i.e., a customer who belongs to one of the classes
{1; 2; : : : ; p}
equal-priority customer, i.e., a customer belonging to class-p.
low-priority customer, i.e., a customer who belongs to one of the classes {p + 1;
p + 2; : : : ; r}
busy period initiation point
the Laplace–Stieltjes transform (LST) of the distribution function (DF) of the waiting
time of a class-p test customer
mean waiting time of a class-p test customer

arrival rate of class-p customers
total arrival rate
arrival rate of EHCs
arrival rate of LCs
service time of a class-p customer (random variable)
= Pr(Sp 6x); DF of Sp
the P
LST of Bp (x)
r
∗
=
k=1 (k =)Bk (); the LST of the DF of the service time of a randomly selected
customer from all classes

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

Bp+∗ ()
Bp−∗ ()
bp
b

b(2)
p
b+
p
b−
p
p bp
p =P
r
= P
k=1 k
p
+
p = Pk=1 k
r
−
p = k=p+1 k
+
p ()
(i1 ; i2 ; : : : ; ir )

R(i1 ; i2 ;:::; ir )

37

Pp
= Pk=1 (k =p+ )Bk∗ (); the LST of the DF of the service time of a randomly selected EHC
r
= k=p+1 (k =p− )Bk∗ (); the LST of the DF of the service time of a randomly selected
LC
=P
E(Sp ); the mean service time of a class-p customer.
r
= k=1 (k =)bk ; the mean service time of a randomly selected customer from all classes
2
= E(S
Ppp ); the +second moment of the service time of a class-p customer
= Pk=1 (k =p )bk ; the mean service time of a randomly selected EHC
r
= k=p+1 (k =p− )bk ; the mean service time of a randomly selected LC
steady-state probability that the server is serving a class-p customer

steady-state probability that the server is busy
steady-state probability that the server is serving an EHC
steady-state probability that the server is serving an LC
= Bp+∗ [ + p+ − p+ +
p ()]; the delay cycle generated by the service of an EHC and its
EHC-osprings
a system state during the idle period in which there are ik class-k customers, (06ik 6Nk −
1; k = 1; 2; : : : ; r)
the remaining idle period from an arbitrary time point in the state (i1 ; i2 ; : : : ; ir ). From
PASTA (Wol [13]), this is equal to the remaining idle period under a condition where
an arriving customer comes upon the state (i1 ; i2 ; : : : ; ir ).

3. Analysis of the idle period
Before we derive the LST of the waiting time distribution of a class-p test customer, it is necessary to
analyze the idle period.
First, the probability that the idle period visits state (i1 ; i2 ; : : : ; ir ) is given by
(i1 ; i2 ;:::; ir ) =

(i1 + i2 + · · · + ir )!
i1 !i2 ! : : : ir !



1


i1 

2


i2

···



r


ir

;

(06ik 6Nk − 1; k = 1; 2; : : : ; r):

(1)

The mean staying time for the state (i1 ; i2 ; : : : ; ir ) is 1= for any combination of i1 ; i2 ; : : : ; ir . Thus, the
steady-state probability that the system is in the state (i1 ; i2 ; : : : ; ir ) under the condition that the system is idle
is given by
(i ; i ;:::; i )
:
P(i1 ; i2 ;:::; ir ) = PN1 −1 PN2 −1 1 2 PrNr −1
i1 =0
i2 =0 · · ·
ir =0 (i1 ; i2 ;:::; ir )

(2)

From PASTA, it is also the probability that a test customer that arrives during the idle period encounters
the state (i1 ; i2 ; : : : ; ir ).
Now, let us obtain the LST of the DF of R(i1 ; i2 ;:::; ir ) . Under the event Aj that the rst arrival during
(i1 ; i2 ; : : : ; ir ) is of class-j, we have
R(i1 ; i2 ;:::; ir ) = Tj + R(i1 ;:::; ij +1;:::; ir ) ;

(3)

where Tj is the time until
Pr the rst arrival under Aj . It is easily seen that Tj follows the exponential distribution
with mean E(Tj )=1= j=1 j =1=. Also, we see that Tj and R(i1 ;:::; ij +1;:::; ir ) are independent. Using Pr(Aj )=j =,

38

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

the LST of the DF of R(i1 ; i2 ;:::; ir ) is given by
R∗(i1 ; i2 ;:::; ir ) () =

r
X
j
j=1



Tj∗ ()R∗(i1 ;:::; ij +1;:::; ir ) ();

(4a)

where
R∗(i1 ;:::; Nk −1;:::; ir ) () =

k−1
X
j ∗
k
Tj ()R∗(i1 ;:::; ij +1;:::; Nk −1;:::; ir ) () + Tk∗ ()


j=1

+

r
X
j ∗
T ()R∗(i1 ;:::; Nk −1;:::; ij +1;:::; ir ) ()
 j

(4b)

j=k+1

and
R∗(N1 −1; N2 −1;:::; Nr −1) () =

r
X
j ∗
T ():
 j

(4c)

j=1

The mean can be obtained recursively from


r
X
j 1
+ E(R(i1 ;:::; ij +1;:::; ir ) ) ;
E(R(i1 ; i2 ;:::; ir ) ) =
 

(5a)

j=1

where

   
k−1
X
j 1
1
k
+ E(R(i1 ;:::; ij +1;:::; Nk −1;:::; ir ) ) +
E(R(i1 ;:::; Nk −1;:::; ir ) ) =
 


j=1



r
X
j 1
+ E(R(i1 ;:::; Nk −1;:::; ij +1;:::; ir ) )
+
 

(5b)

j=k+1

and
1
E(R(N1 −1; N2 −1;:::; Nr −1) ) = :


(5c)

4. Waiting times under FCFS
In this section, we derive the LST of the distribution function
Pr of the waiting time of a∗class-p test customer
under FCFS. The server is busy with a probability of  = k=1 k . Thus, the LST Wq(p);
FCFS () of the DF
of the waiting time of a class-p test customer can be written as
∗
∗
∗
Wq(p);
FCFS () = Wq(p); FCFS ( | busy) + (1 − )Wq(p); FCFS ( | idle):

(6)

∗
To obtain Wq(p);
FCFS ( | busy) we need to nd the probability distribution for the number of customers at
the busy period initiation point (BPIP). Let K(z1 ; z2 ; : : : ; zr ) be the joint PGF (probability generating function)
for the number of customers of each class at the BPIP. After laborious manipulations, we get (a derivation

39

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

of (7) is provided in the appendix).
K(z1 ; z2 ; : : : ; zr )
=

N1



1 z1


+



 r zr


Nr



 1 z1


i1

NX
r −1

×

···

ir =0

NX
2 −1
i2 =0

Nr−1 −1

X

×

···

ir−1 =0

···

(i2 + · · · + ir )!
i2 ! : : : ir !



NX
1 −1
i1 =0

r−1 zr−1




(i1 + · · · + ir−1 )!
i1 ! : : : ir−1 !

ir−1

2 z2


i2

Nr + i1 + · · · + ir−1 − 1
i1 + · · · + ir−1



N1 + i2 + · · · + ir − 1
i2 + · · · + ir




···



r zr


ir

+ ···

(7)

:

Let T0 be the time that it takes to serve all existing customers at the BPIP. Let T0∗ () be the LST of the
DF of T0 . Then T0∗ () can be obtained from
T0∗ () = K(z1 ; z2 ; : : : ; zr )|z1 =B1∗ ();z2 =B2∗ ();:::; zr =Br∗ () :

(8a)

The rst moment becomes
d
E(T0 ) = − T0∗ ()|=0
d

 N1 NX
  i2
NX
r −1
2 −1
1
(i2 + · · · + ir )! N1 + i2 + · · · + ir − 1
2
=
×
···
···
i2 + · · · + ir

i2 ! : : : ir !

ir =0

+

ir

i2 =0

×



r




p


Np

×



N1 + i1 + · · · + ip−1 + ip+1 + · · · + ir − 1
i1 + · · · + ip−1 + ip+1 + · · · + ir

×



1


[N1 b1 + i2 b2 + · · · + ir br ] + · · ·
NX
r −1

×

Np+1 −1 Np−1 −1

···

ir =0

i1

···



X

X

···

ip+1 =0 ip−1 =0

p−1


ip−1 

p+1


NX
1 −1
i1 =0

ip+1

(i2 + · · · + ip−1 + ip+1 + · · · + ir )!
i2 ! : : : ip−1 !ip+1 ! : : : ir !

···



r




ir

×[Np bp + i1 b1 + · · · + ip−1 bp−1 + ip+1 bp+1 + · · · + ir br ] + · · ·
+

Nr



r




r−1


Nr−1 −1

×

X

ir−1 =0

ir−1

i1 =0

(i1 + · · · + ir−1 )!
i1 ! · · · ir−1 !

[Nr br + i1 b1 + · · · + ir−1 br−1 ]:

The second moment becomes
E(T02 ) =

···

NX
1 −1

d2 ∗
T ()|=0
d 2 0



Nr + i1 + · · · + ir−1 − 1
i1 + · · · + ir−1



1


i1

···

(8b)

40

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

=



1


N1

NX
r −1

×

···

ir =0

(

i2 =0

1)b21

× N1 (N1 −

NX
2 −1

+

N1 b(2)
1

+

r
X



[ia (ia −

N1 + i2 + · · · + ir − 1
i2 + · · · + ir
1)b2a

ia b(2)
a ]

+

+2

a=2

+ 2N1 b1

r
X

ia ba

a=2

+

(i2 + · · · + ir )!
i2 ! : : : ir !

NX
r −1

)

2


" r−1 r
X X

i2

···



r


ia ic ba bc −

a=1 c=a+1

ir
r
X

i1 ic b1 bc

c=2

#

+ ···
Np+1 −1 Np−1 −1

NX
1 −1



p


Np

×



N1 + i1 + · · · + ip−1 + ip+1 + · · · + ir − 1
i1 + · · · + ip−1 + ip+1 + · · · + ir

×



ir =0

X

X

···

···

ip+1 =0 ip−1 =0

i1 =0

(i2 + · · · + ip−1 + ip+1 + · · · + ir )!
i2 ! : : : ip−1 !ip+1 ! : : : ir !


1


i1

···



p−1


ip−1 

p+1


ip+1

···

i
r r



r
X


2
(2)
2
(2)




Np (Np − 1)bp + Np bp +
[ia (ia − 1)ba + ia ba ]








a=2


a6=k
#
"
×
+ ···
r−1
r
r
k−1
r
X X
X
X
X






i
i
b
b
−
i
i
b
b
−
b
i
b
i
i
b
b
+
2N
+2

a c a c
k c k c
k k
a a
a k a k






a=1 c=a+1
a=1
a=1
c=k+1





a6=k

+



r


Nr

×



1


Nr−1 −1

×

i1

X

···

ir−1 =0

···

(

× Nr (Nr −

+ 2Nr br



i1 =0

r−1


1)b2r

NX
1 −1

+

(i1 + · · · + ir−1 )!
i1 ! : : : ir−1 !

a=1

ia ba

Nr + i1 + · · · + ir−1 − 1
i1 + · · · + ir−1



ir−1
Nr b(2)
r

+

r−1
X

[ia (ia −

a=1

r−1
X



)

:

1)b2a

+

ia b(2)
a ]

+2

" r−1 r
X X

a=1 c=a+1

ia ic ba bc −

r−1
X
a=1

ia ir ba br

#

(8c)

The busy period is a delay cycle with T0 as the initial delay. The LST of the distribution function of the
waiting time of an arbitrary customer that arrives during such a delay cycle under FCFS is given by (see
Takagi [12])
∗
Wq(p);
FCFS ( | busy) =

(1 − )
1 − T0∗ ()
E(T0 )  −  + B∗ ()

(9)

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

41

Fig. 1. Mean waiting time when N1 varies (FCFS).

with mean
E(Wq(p); FCFS | busy) =

b(2)
E(T02 )
+
:
2E(T0 ) 2(1 − )

(10)

Under the FCFS discipline, the DFs of the waiting times of the customers that arrive during the busy period
do not dier among classes. Thus (9) and (10) can be applied to all classes.
Now let us obtain the LST of the DF of the waiting time of the class-p test customer that arrives during
the idle period. Let E(i1 ;:::; ir )(p) be the event that the class-p test customer encounters an idle state (i1 ; : : : ; ir )
when it arrives. Let w(i∗1 ;:::; ir )(p) () be the LST of the DF of the waiting time under E(i1 ;:::; ir )(p) . Then we get
w(i∗1 ;:::; ir )(p) () = [B1∗ ()]i1 : : : [Br∗ ()]ir R∗(i1 ;:::; ip +1;:::ir ) ():

(11)

From PASTA, we have
∗
Wq(p);
FCFS ( | idle) =

NX
1 −1

···

i1 =0

NX
r −1

P(i1 ;:::; ir ) w(i∗1 ;:::; ir )(p) ();

(12)

ir =0

∗
∗
where P(i1 ;:::; ir ) was obtained in (2). Unlike Wq(p);
FCFS ( | busy); Wq(p); FCFS ( | idle) depends on p. It can be
shown that the mean becomes

E(Wq(p); FCFS | idle) =

NX
1 −1

···

i1 =0

NX
r −1

P(i1 ;:::; ir ) [i1 b1 + · · · + ir br + E(R(i1 ;:::; ip +1;:::; ir ) )]:

(13)

ir =0

Using (9) and (12) in (6), we get
∗
Wq(p);
FCFS () = 

NX
NX
r −1
1 −1
(1 − )
1 − T0∗ ()
+
(1
−
)
P(i1 ;:::; ir ) w(i∗1 ;:::; ir )(p) ():
·
·
·
E(T0 )  −  + B∗ ()
i1 =0

(14)

ir =0

The mean waiting time becomes
E(Wq(p); FCFS ) = E(Wq(p); FCFS | busy) + (1 − )E(Wq(p); FCFS | idle)


NX
NX
1 −1
r −1
E(T02 )
b(2)
+ (1 − )
···
P(i1 ;:::; ir ) [i1 b1 + · · · + ir br + E(R(i1 ;:::; ip +1;:::; ir ) )]:
=
+
2E(T0 ) 2(1 − )
i1 =0

ir =0

(15)
The mean number of customers can be obtained from the Little’s law.
Figs. 1 and 2 show the behavior of the mean waiting time when one of the thresholds varies while the
other two are xed.

42

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

Fig. 2. Mean waiting time when N3 varies (FCFS).

5. Waiting times under non-preemptive priority
In this section, we consider the waiting times in the case of non-preemptive priority (NP). We still have
∗
∗
∗
Wq(p);
NP () = Wq(p); NP ( | busy) + (1 − )Wq(p); NP ( | idle):

(16)

First, we obtain X (z), the PGF of the number of EHCs at the BPIP. This can be obtained by using z in
place of z1 ; z2 ; : : : zp and by using 1 in zp+1 ; zp+2 ; : : : ; zr in (7):
X (z) = K(z1 ; z2 ; : : : ; zr )|z1 =z2 =···=zp =z;

(17)

zp+1 =zp+2 =···=zr =1 :

∗
Let T0(p) be the time that it takes to serve all EHCs existing at the BPIP and T0(p)
() be its LST. Then
we get
∗
() = X [Bp+∗ ()]:
T0(p)

(18a)

The rst moment becomes
d ∗
T ()|=0
d 0(p)

  i2
 N1 NX
NX
r −1
2 −1
2
1
(i2 + · · · + ir )! N1 + i2 + · · · + ir − 1
=
×
···
···
+
·
·
·
+
i
i

i2 ! : : : ir !

2
r

E(T0(p) ) = −

ir =0

+

i2 =0



r


ir



p


Np

×



N1 + i1 + · · · + ip−1 + ip+1 + · · · + ir − 1
i1 + · · · + ip−1 + ip+1 + · · · + ir

×



1


(N1 + i2 + · · · + ip )b+
p + ···

×

i1

NX
r −1
ir =0

···



Np+1 −1 Np−1 −1

···

X

X

···

ip+1 =0 ip−1 =0

p−1


ip−1 

p+1


NX
1 −1
i1 =0

ip+1

(i2 + · · · + ip−1 + ip+1 + · · · + ir )!
i2 ! : : : ip−1 !ip+1 ! : : : ir !

···



r




ir

(Np + i1 + · · · + ip−1 )b+
p

43

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

+

+

Np+1

NX
r −1

Np+2 −1 Np −1

NX
1 −1



p+1


×



N1 + i1 + · · · + ip + ip+2 + · · · + ir − 1
i1 + · · · + ip + ip+2 + · · · + ir

×



1




r


Nr

×



1


i1

×

ir =0

···



p


X

ip 

···

ir−1 =0

i1

:::



p+2


NX
1 −1
i1 =0

r−1


···

ip+2 =0 ip =0

Nr−1 −1

×

X X

···

ir−1

ip+2

i1 =0

···



r


(i2 + · · · + ip + ip+2 + · · · + ir )!
i2 ! : : : ip !ip+2 ! : : : ir !

ir

(i1 + · · · + ir−1 )!
i1 ! : : : ir−1 !


(i1 + · · · + ip )b+
p + ···



Nr + i1 + · · · + ir−1 − 1
i1 + · · · + ir−1



(i1 + · · · + ip )b+
p:

(18b)

The second moment becomes
d2 ∗
T ()|=0
d 2 0(p)

 N1 NX
 ir
  i2
NX
r −1
2 −1
1
r
(i2 + · · · + ir )! N1 + i2 + · · · + ir − 1
2
=
×
···
···
+
·
·
·
+
i
i

i2 ! : : : ir !


2
r

2
)=
E(T0(p)

ir =0

i2 =0

2
+(2)
×[(N1 + i2 + · · · + ip )(N1 + i2 + · · · + ip − 1)(b+
p ) + (N1 + i2 + · · · + ip )bp ] + · · ·

+

NX
r −1

Np+1 −1 Np−1 −1

NX
1 −1



p


Np

×



N1 + i1 + · · · + ip−1 + ip+1 + · · · + ir − 1
i1 + · · · + ip−1 + ip+1 + · · · + ir

×



1


×

i1

···

ir =0

···



X

X

···

ip+1 =0 ip−1 =0

p−1


ip−1 

p+1


i1 =0

ip+1

(i2 + · · · + ip−1 + ip+1 + · · · + ir )!
i2 ! : : : ip−1 !ip+1 ! : : : ir !

···



r




ir

2
×[(N1 + i1 + i2 + · · · + ip−1 )(N1 + i1 + i2 + · · · + ip−1 − 1)(b+
p)

+(Np + i1 + · · · + ip−1 )b+(2)
p ]
+

Np+1

NX
r −1

Np+2 −1 Np −1

NX
1 −1



p+1


×



N1 + i1 + · · · + ip + ip+2 + · · · + ir − 1
i1 + · · · + ip + ip+2 + · · · + ir

×



1


i1

×

···

···

ir =0



p


ip 

X X

···

ip+2 =0 ip =0

p+2


ip+2

i1 =0

···



r


(i2 + · · · + ip + ip+2 + · · · + ir )!
i2 ! : : : ip !ip+2 ! : : : ir !

ir



44

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49
2
+(2)
×[(i1 + · · · + ip )(i1 + · · · + ip − 1)(b+
p ) + (i1 + · · · + ip )bp ] + · · ·

+



r


Nr

×



1


Nr−1 −1

X

×

NX
1 −1

···

ir−1 =0

i1

···



r−1


i1 =0

(i1 + · · · + ir−1 )!
i1 ! : : : ir−1 !



Nr + i1 + · · · + ir−1 − 1
i1 + · · · + ir−1



ir−1

2
+(2)
×[(i1 + · · · + ip )(i1 + · · · + ip − 1)(b+
p ) + (i1 + · · · + ip )bp ]:

(18c)

The busy period is composed of two types of intervals:
(i) T0(p) -cycle: The time duration from the BPIP to the rst time when there are no EHC s. It is a delay
cycle which has T0(p) as the initial delay and is extended by EHCs and their EHC-osprings.
(ii) Bp− -cycle: A delay cycle that starts with the service of an LC (there are no EHC s at this point) and is
extended by its EHC-osprings.
Let T0(p) and Bp− be the probabilities of the class-p test customer that arrives during the busy period and
nds T0(p) -cycle and Bp− -cycle, respectively. Since every LC generates its own Bp− -cycle, we get, after using
!
b−
p
−
;
E(Bp -cycle) =
1 − +
p
Bp− =

p−

b−
p
1 − +
p

!

=

−
p
:
1 − +
p

(19)

From T0(p) + Bp− = , we get
T0(p) =

+
p (1 − )
:
1 − +
p

(20)

We can apply the same delay-cycle argument that we employed in the FCFS case (Eq. (9)) and get
∗
Wq(p);
NP ( | busy) =

+
∗
−∗ +
Bp− (1 − +
T0(p) (1 − +
p )[1 − T0(p) (p−1 )]
p )[1 − Bp (p−1 )]
;
+
+
+
 E(T0(p) )[ − p + p Bp∗ (p−1
 E(Bp− )[ − p + p Bp∗ (p−1
)]
)]

(21)
+
p−1

+
=  + p−1

+
− p−1
+∗
p−1 ()

+∗
+
+∗
p−1 () = Bp−1 (p−1 ).

and
Note that
where
starts with the service of a HC and is extended by its HC-osprings.
The mean becomes
!
2
E(T0(p)
) p+ b+(2)
(1 − )+
p
p
+
E(Wq(p); NP | busy) =
+
E(T0(p) )
1 − +
2(1 − +
p
p−1 )(1 − p )
+

−
p
+
2(1 − +
p−1 )(1 − p )

p+ b+(2)
b−(2)
p
p
+
1
−
+
b−
p
p

!

:

+∗
p−1 ()

is a busy period that

(22)

The waiting time of a class-p test customer that arrives during the idle period consists of
(i) the remaining idle period,
(ii) the time it takes to service all HC s that arrive during the entire idle period and their HC-osprings, and

45

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

(iii) the time it takes to serve all EC s that already exist at the arrival point of the test customer and their
EC-osprings.
Let (i1 ; i2 ;:::; ir )(p) be the sum of (i) and (ii) under the condition that the arriving class-p test customer encounters
an idle state (i1 ; i2 ; : : : ; ir ). Let (i∗ 1 ; i2 ;:::; ir )(p) () be the LST of the DF of (i1 ; i2 ;:::; ir )(p) . For p = 1, (ii) becomes
zero and (i∗ 1 ; i2 ;:::; ir )(1) () is equal to R∗(i1 ; i2 ;:::; ir ) (). Conditioning on the class of the rst arriving customer, we
get
(i∗ 1 ; i2 ;:::; ir )(p) () =

r
X
j
j=1



Tj∗ ()(i∗ 1 ; i2 ;:::; ij +1;:::; ir )(p) ();

(23a)

where
+∗
+
(p−1
)]i1 +···+ip−1 :
(i∗ 1 ;:::; Nj ;:::; ir )(p) () = [Bp−1

Tj∗ ()

(23b)

was dened and used in (4a). The mean becomes


r
X
j 1
+ E((i1 ;:::; ij+1 ;:::; ir )(p) ) ;
E((i1 ; i2 ;:::; ir )(p) ) =
 

(24a)

j=1

where

E((i1 ;:::; Nj ;:::; ir )(p) ) =





b+

p−1



 1 − +
p−1




b+

p−1


 1 − +
p−1





p−1
X 

Nj +
ik 

 ; (j = 1; 2; : : : ; p − 1);
k=1

p−1

X

ik

k=1

!(k6=j)
;

(24b)

(j = p; p + 1; : : : ; r):

Let w(i∗1 ;:::; ir )(p) () be the LST of the DF of the waiting time of the class-p test customer that arrives during
the idle period and encounters the state (i1 ; : : : ; ir ). w(i∗1 ;:::; ir )(p) () is the LST of the DF of (i) + (ii) + (iii) and
is given by
+
w(i∗1 ;:::; ir )(p) () = (i∗ 1 ;:::; ip +1;:::; ir )(p) ()[Bp∗ (p−1
)]ip :

(25)

Then we get
∗
Wq(p);
NP ( | idle) =

NX
1 −1

···

i1 =0

The mean becomes
E(Wq(p); NP | idle) =

NX
1 −1
i1 =0

NX
r −1

P(i1 ;:::; ir ) w(i∗1 ;:::; ir )(p) ():

(26)

ir =0

···

NX
r −1

P(i1 ;:::; ir )

ir =0

"

#
ip bp
+ E((i1 ;:::; ip+1; :::; ir )(p) ) :
1 − +
p−1

(27)

∗
Finally, Wq(p);
NP () can be obtained by using (21) and (26) in (16), and the mean waiting time becomes

E(Wq(p); NP ) = E(Wq(p); NP | busy) + (1 − )E(Wq(p); NP | idle)
"
!
2
E(T0(p)
) p+ b+(2)
(1 − )+
p
p
=
+
+
E(T0(p) )
1 − +
2(1 − +
p
p−1 )(1 − p )
+

−
p
+
2(1 − +
p−1 )(1 − p )

p+ b+(2)
b−(2)
p
p
+
−
1 − +
bp
p

!#

46

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

Fig. 3. Mean waiting time when N1 varies (NP).

Fig. 4. Mean waiting time when N3 varies (NP).

+ (1 − )

NX
1 −1
i1 =0

···

NX
r −1
ir =0

P(i1 ; i2 ;:::; ir )

"

#
ip bp
+ E((i1 ;:::; ip +1;:::; ir )(p) :
1 − +
p−1

(28)

The mean number of customers can be obtained from the Little’s law.
Figs. 3 and 4 show the behavior of the mean waiting time when one of the thresholds varies while the
other two are xed.

Acknowledgements
The authors are very thankful to anonymous referees for their helpful comments. Their careful readings
greatly enhanced the richness and readability of this paper.

47

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

Appendix Derivation of Eq. (7)
Let us consider the case for three classes. The r-class case is a simple extension. Let TNp be the time until
N
Np class-p customers arrive. TNp follows the Erlang distribution with pdf fNp (t) = pp t Np −1 e−p t =(Np − 1)!.
∗
Let FNp (t) be the distribution function of TNp and let FNp () be the LST of FNp (t). Then we have

K(z1 ; z2 ; z3 ) =

NX
1 −1 N
2 −1
X

∞

(2 t)m e−2 t (3 t)m e−3 t
dFN1 (t)
m!
n!

z1m z2N2 z3n

Z

∞

(1 t)m e−1 t (3 t)m e−3 t
dFN2 (t)
m!
n!

z1m z2n z3N3

Z

∞

(1 t)m e−1 t (1 t)m e−1 t
dFN3 (t):
m!
n!

z1N1 z2m z3n

n=0 m=0

+

NX
3 −1 N
1 −1
X

Z

0

n=0 m=0

+

NX
2 −1 N
1 −1
X
n=0 m=0

0

0

(A.1)

It suces to see the rst term on the right-hand side of (A.1). It becomes
NX
3 −1 N
2 −1
X

z1N1 z2m z3n

n=0 m=0

=

NX
3 −1 N
2 −1
X

Z

∞

0

z1N1 z2m z3n

n=0 m=0

(2 t)m e−2 t (3 t)m e−3 t
·
dFN1 (t)
m!
n!
1 1
(2 )m (3 )n
m! n!

Z

∞

t n+m e−(2 +3 )t dFN1 (t):

(A.2)

0

To evaluate the right-hand side of (A.2), we consider the following identity:
Z

∞

k
dk ∗
k d
F
()
=
(−1)
t dFN1 (t) = (−1)
N
dk 1
dk

−t k

e
0

=



1
1 + 

k

N1 

1
1 + 



1
1 + 

N1

k


k!


[N1 (N1 − 1) · · · (N1 − k + 1)] +
[N1 (N1 − 1) · · · (N1 − k + 2)]


(k − 2)!1!





(k − 1)k!



 + (k − 3)!2! [N1 (N1 − 1) · · · (N1 − k + 3)]






























(k − 1)(k − 2) · · · 3k!
(k − 1)(k − 2) · · · 4k!

[N1 (N1 − 1)(N1 − 2)] +
[N1 (N1 − 1)] 


2!(k − 3)!
1!(k − 2)!





(k − 1)(k − 2) · · · 2k!



N1
+
(k − 1)!
+ ··· +

:

(A.3)

48

H.W. Lee et al. / Operations Research Letters 28 (2001) 35–49

By replacing  by 2 + 3 and k by n + m in (A.3), (A.2) becomes
Z ∞
NX
3 −1 N
2 −1
X
(2 t)m e−2 t (3 t)m e−3 t
z1N1 z2m z3n
·
dFN1 (t)
m!
n!
0
n=0 m=0

=

NX
3 −1
2 −1 N
X

z1N1 z2m z3n

n=0 m=0

where

1 1
(2 )m (3 )n
m! n!



1


N1  n+m
1
× B;




(m + n)!



[N1 (N1 − 1) · · · (N1 − m − n + 2) 
[N1 (N1 − 1) · · · (N1 − m − n + 1)] +




(m
+
n
−
2)!1!






(m + n − 1)(m + n)!






[N
(N
−
1)
·
·
·
(N
−
m
−
n
+
3)
+
·
·
·
+
1
1
1




(m
+
n
−
3)!2!



 (m + n − 1)(m + n) · · · 4 · (m + n)!
[N1 (N1 − 1)(N1 − 2)]
:
B= +


2!(m + n − 3)!






(m + n − 1)(m + n − 2) · · · 3 · (m + n)!




[N1 (N1 − 1)]



+
1!(m
+
n
−
2)!






(m
+
n
−
1)(m
+
n
−
2)
·
·
·
2
·
(m
+
n)!




N1

+
(m + n − 1)!

Now using the identity
  
 

n
n−1
n−1
=
+
;
r
r−1
r

we get

B = (m + n)!

m+n−1
X 
j=0

m+n−1
j



N1
j+1



= (m + n)!



N1 + m + n − 1
m+n



:

Thus (A.2) becomes
NX
3 −1 N
2 −1
X

z1N1 z2m z3n

n=0 m=0

=

NX
3 −1
2 −1 N
X
n=0 m=0

=



1 z1


Z

∞

0

z1N1 z2m z3n

(2 t)m e−2 t (3 t)m e−3 t
dFN1 (t)
m!
n!
1 1
(2 )m (3 )n
m! n!

N1 NX
3 −1 
2 −1 N
X
n=0 m=0

m+n
m





1


N1  n+m
1
×B


N1 + m + n − 1
m+n



2 z2


m 

3 z3


n

:

Similar derivations can be applied to the second and third terms of (A.1) to get (7).
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