Network of M/M/1 Cyclic Polling Systems
Carlos Mart
´
ınez-Rodr
´
ıguez
1
, Ra
´
ul Montes-de-Oca
2
and Patricia Saavedra
2
1
Universidad Aut
´
onoma de la Ciudad de M
´
exico, Calzada Ermita Iztapalapa 4163,
Col. Lomas de Zaragoza, 09620, Ciudad de M
´
exico, Mexico
2
Universidad Aut
´
onoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186,
Col. Vicentina, 09340, Ciudad de M
´
exico, Mexico
Keywords:
Networks of Cyclic Polling System, Exhaustive Policy, Exponential Inter-arrivals Times.
Abstract:
This paper presents a Network of Cyclic Polling Systems that consists of two cyclic polling systems with
two queues each when transfer of users from one system to the other is imposed. This system is modelled
in discrete time. It is assumed that each system has exponential inter-arrival times and the servers apply an
exhaustive policy. Closed form expressions are obtained for the first and second moments of the queue’s
lengths for any time.
1 INTRODUCTION
A Cyclic Polling System (CPS) consists of multiple
queues that are attended by a single server in cyclic
order. Users arrive at each queue according to inde-
pendent processes which are independent of the ser-
vice times. The server attends each queue accord-
ing to a service policy previously established. When
the server finishes, it moves to the next queue incur-
ring in a switchover time. It will be assumed that
the switchover times form a sequence of independent
and identically distributed random variables. A thor-
ough analysis has been made on this subject. For an
overview of the literature on polling systems, their ap-
plications and standard results, the authors refer to
such surveys as: (Boon et al., 2011; Levy and Sidi,
1990), and (Vishnevskii and Semenova, 2006).
Here a Network of Cyclic Polling System (NCPS)
is considered. It consists of two cyclic polling sys-
tems, each of them with two queues that are attended,
according to an exhaustive policy. The exhaustive
policy service consists in attending all users until the
queue is emptied. The system is observed at fixed
times where the length of the slot is proportional to
the time service. The arrivals to each queue are as-
sumed to be Poisson processes with independent iden-
tical distributed (i.i.d.) inter-arrival exponential times.
When the servers finish, they move to the next queue
incurring a switchover time. It will be assumed that
the switchover times form a sequence of indepen-
dent and identically distributed random variables. The
novelty in this work is that the two systems are con-
nected in the following way: the users enter the sys-
tem through one of the queues. After being served in-
stead of leaving the system, they transfer to one of the
queues of the other system, see Figure 1. All the users
leave the network after being attended by the two
servers. This network requires considering two kinds
of arrival processes at each queue. One of them cor-
responds to the arrival process of the users that enter
the system for the first time through that queue, and
the other one corresponds to the arrival of the transfer
users. Specifically, in this article the authors are look-
ing for explicit formulae for the first and second order
moments at any time. The buffer occupancy method
is applied. It uses the Probability Generating Function
(PGF) of the joint distribution function of the queues
lengths at the moment the server arrives to the queue
to start its service, which is called a polling instant.
For an overview of this method, see (Takagi, 1986;
Cooper and Murray, 1969; Cooper, 1970).
This work was motivated by the subway system,
where each line can be considered as a cyclic polling
system and the transfer station allows the users to
transfer from one line to the other. Networks of
polling systems is a rather new topic with few ref-
erences, and a variety of possible applications, see
(Boon et al., 2011; Levy and Sidi, 1990; Vishnevskii
and Semenova, 2006; Beekhuizen, 2010). Recent
publications about networks of polling stations are:
(Beekhuizen et al., 2008b; Beekhuizen et al., 2008a;
Aoun et al., 2010; Beekhuizen and Resing, 2009;
298
Martà nez-Rodrà guez C., Montes-de-Oca R. and Saavedra P.
Network of M/M/1 Cyclic Polling Systems.
DOI: 10.5220/0006143802980305
In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 298-305
ISBN: 978-989-758-218-9
Copyright
c
2017 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
Figure 1: Example of a Network of Cyclic Polling Systems.
van den Bos and Boon, 2013). The problem of in-
terest is to obtain performance measures for the sta-
tionary case whenever is possible.
This paper is organized as follows. In section 2 the
description of the model and the corresponding nota-
tion are presented. In Section 3 the explicit formulae
for the queue length processes at polling instants are
given. Assuming that stationarity conditions are satis-
fied, the expected queue lengths processes at any time
are provided in Section 4. The concluding remarks
are given in section 5. Besides there is Appendix A,
which gives the general calculations in order to obtain
the joint PGF for the queue lengths of the NCPS.
2 DESCRIPTION OF THE MODEL
Consider an NCPS consisting of two cyclic polling
systems, Γ
1
and Γ
2
with two queues each, where Q
1
and Q
2
denote the queues of Γ
1
, and Q
3
and Q
4
de-
note the corresponding queues of Γ
2
, all of them with
infinite-sized buffer. In each system, a single server
visits the queues in a cyclic order, where the exhaus-
tive policy is applied. All the users after being served,
transfer to the other system in the following way:
users from Q
3
transfer to Q
1
, and from Q
4
to Q
2
, and
viceversa. Users’ time of arrival to the other queue is
considered as the time of departure from the original
queue. All customers are assumed to leave the NCPS
after being attended by the two servers, see Figure 1.
Upon completion of a visit to any queue, the
servers incur in a random switchover time according
to an arbitrary distribution with a finite first moment.
A cycle is defined as the time interval between two
consecutive polling instants. The time period in a cy-
cle during which the server attends a queue is called
a service period. The intervisit time I
i
of queue Q
i
is
defined as the period beginning at the time the server
leaves Q
i
in a cycle and ends at the time when queue
Q
i
is polled in the next cycle; its duration is given
by τ
i
(m + 1) τ
i
(m). It is important to remark that
the case considered in this paper is the one where the
server visits the queues in a cyclic order.
At each of the queues in the network, the total
number of users is the users that arrive for the first
time to the system plus the number of transfer users
from the other system. For t [t,t + 1) the arrival
processes are denoted by X
1
(t),X
2
(t), for Q
1
and Q
2
in Γ
1
, respectively, and X
3
(t), X
4
(t), for Q
3
and Q
4
in Γ
2
, respectively, with corresponding transfer pro-
cesses Y
3
(t) from Q
1
to Q
3
, Y
4
(t) from Q
2
to Q
4
,
Y
1
(t) from Q
3
to Q
1
, and Y
2
(t) from Q
4
to Q
2
. It
will be supposed that the arrival and the transference
processes are independent. The arrival rates at Q
i
,
for i = 1,2,3,4, are denoted by µ
i
and ˆµ
i
for the out-
put processes, respectively. The process that consid-
ers both input-output processes will be denoted by
˜
X
i
(t) = X
i
(t) +Y
i
(t) with a rate ˜µ
i
= µ
i
+ ˆµ
i
that sat-
isfies ˜µ
i
< 1, for i = 1,2,3,4. Denote the processes
L
i
(t) for the queue length processes for i = 1,2,3,4.
In some parts of the article, in order not to complicate
the notation, the dependence of t will be omitted.
3 EXPECTED QUEUE LENGTHS
PROCESSES AT DISCRETE
TIME
In this section it is assumed that the service times are
proportional to the length of the slot, so that the arrival
rate and the output rates coincide with the mean of the
corresponding processes.
As usual, the j-th derivative of a function Ψ is
denoted by Ψ
( j)
, j = 1,2,3,.... When Ψ is a func-
tion of m variables, the notation D
j
Ψ will be used
for the j-th partial derivative of Ψ, j = 1,2, . . . , m.
For i = 1,2,3,4, consider z
i
C and denote by τ
i
the
polling instant at queue Q
i
and by
τ
i
the instant when
the server leaves the queue and starts a switchover
time. In order to obtain the joint PGF for the num-
ber or users residing in queue Q
i
, when it is polled,
for t 0 the PGF is considered for each of the ar-
rival processes X
i
(t), the transfer process Y
i
(t), and
the processes
˜
X
i
(t), for i = 1, 2, 3, 4. The correspond-
ing PGFs for each of the processes are:
P
i
(z
i
) = E
h
z
X
i
(t)
i
i
,
ˆ
P
i
(z
i
) = E
h
z
Y
i
(t)
i
i
, (1)
and
˜
P
i
(z
i
) = E
h
z
˜
X
i
(t)
i
i
, (2)
with
µ
i
= E [X
i
(t)] = P
(1)
i
(1), (3)
Network of M/M/1 Cyclic Polling Systems
299
and ˆµ
i
, ˜µ
i
for the mean of the respective processes
ˆ
Y
i
(t) and
˜
X
i
(t) for i = 1,2,3,4. The PGF for the ser-
vice period is defined by:
S
i
(z
i
) = E
h
z
τ
i
τ
i
i
i
, with
s
i
= E [τ
i
τ
i
], for i = 1,2,3,4.
(4)
In a similar manner, the PGF for the switchover time
of the server from the moment it stops attending a
queue to the time of arrival to the next queue is given
by
R
i
(z
i
) = E
h
z
τ
i+1
τ
i
i
i
, (5)
with the first moment
r
i
= E [τ
i+1
τ
i
] for i = 1,2,3,4. (6)
Observe that the number of users in the queue at times
τ
i
is zero, i.e., L
i
(τ
i
) = 0 for i = 1, 2, 3, 4, and in Γ
1
,
the number of users at the moment the server stops
attending the queue is given by the number of users
present at the moment it arrives plus the number of
arrivals during the service period plus the users that
arrived after being served by the second server. Then
the length L
i
(τ
1
) is given by
L
i
(τ
1
) = L
i
(τ
1
) + X
i
(τ
1
τ
1
) +Y
i
(τ
1
τ
1
), (7)
for i = 1,2,3,4. As it is known, the gambler’s ruin
problem can be used to model the server’s busy pe-
riod in a cyclic polling system. The result that re-
lates the gambler’s ruin problem with the busy pe-
riod of the server is a generalization of the result
given in (Takagi, 1986), Chapter 3. Denote by
˜
L
j
,
j = 0,1,2,..., the capital equal to j units, and by
g
n,k
the probability of the event no ruin before the n-
th period beginning with the initial capital
˜
L
0
, con-
sidering a capital equal to k units after n 1 events,
i.e., given n
{
1,2,...
}
, and k
{
0,1,2,...
}
, g
n,k
:=
P
˜
L
j
> 0, j = 1,...,n,
˜
L
n
= k
. This probability can
be written as:
g
n,k
= P
˜
L
j
> 0, j = 1,...,n,
˜
L
n
= k
=
k+1
j=1
g
n1, j
P
˜
X
n
= k j + 1
=
k+1
j=1
g
n1, j
P
{
X
n
+Y
n
= k j + 1
}
=
k+1
j=1
j
l=1
g
n1, j
P
{
X
n
= k j l + 1
}
·P
{
Y
n
= l
}
.
(8)
Let G
n
(z) and G (z,w) be the polynomials defined by
G
n
(z) =
k=0
g
n,k
z
k
, for n = 0,1,..., and
G(z,w) =
n=0
G
n
(z)w
n
,
(9)
for z,w C, where it is obtained that
g
0,k
= P
˜
L
0
= k
. (10)
In particular for k = 0,
g
n,0
= G
n
(0) = P
˜
L
j
> 0,
˜
L
n
= 0
= P
{
T = n
}
,
(11)
for j < n and the ruin time T . Furthermore,
G(0,w) =
n=0
G
n
(0)w
n
=
n=0
P
{
T = n
}
w
n
= E
w
T
,
(12)
is the PGF of T . The gambler’s ruin occurs after
the n-th game, i.e., the queue becomes empty after n
steps, starting with
˜
L
0
users.
Proposition 1. For n 0, z,w C, z 6= 0,
G
n
(z) =
1
z
[G
n1
(z) G
n1
(0)]
˜
P
i
(z).
Furthermore,
G(z,w) =
zF
i
(z) w
˜
P
i
(z)G(0,w)
z wR
i
(z)
, (13)
z wR
i
(z) 6= 0, with a unique pole in the unit circle,
which has the form z =
˜
θ
i
(w) and satisfies
i)
˜
θ
i
(1) = 1,
ii)
˜
θ
(1)
i
(1) = 1/[1 ˜µ
i
],
iii)
˜
θ
(2)
i
(1) = ˜µ
i
/(1 ˜µ
i
)
2
+
˜
σ/(1 ˜µ
i
)
3
,
for i = 1,2,3,4.
Proof. Similar to the one given by Takagi (Takagi,
1986) in pp. 45-47.
In order to model the NCPS it is necessary to con-
sider the users arrival to each queue of Γ
1
at polling
instants of system Γ
2
. The PGF of the queue length
of system Γ
1
at polling instants of Γ
2
is defined as
F
i,i+2
(z
i
;τ
i+2
) = E
h
z
L
i
(τ
i+2
)
i
i
, (14)
for z
i
C,i = 1,2,3,4. Using this expression, it is
possible to define the joint PGF for Γ
1
, z
1
,z
2
C:
E
h
z
L
1
(τ
3
)
1
z
L
2
(τ
3
)
2
i
= E
h
z
L
1
(τ
3
)
1
i
E
h
z
L
2
(τ
3
)
2
i
= F
1,3
(z
1
;τ
3
)F
2,3
(z
2
;τ
3
) =: F
3
(z
1
,z
2
;τ
3
).
(15)
Similar expressions are obtained for the rest of the
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
300
queues which can be summarized by
F
j
(z
1
,z
2
;τ
j
), for j = 3, 4 and
F
j
(z
3
,z
4
;τ
j
), for j = 1, 2,
(16)
for z
i
C, i = 1,2,3,4. Now the joint PGF will be de-
termined for the times that the servers visit each queue
in their corresponding system, i.e., t
{
τ
1
,τ
2
,τ
3
,τ
4
}
:
F
j
:= F
j
(z
1
,z
2
,z
3
,z
4
;τ
j
) = E
4
i=1
z
L
i
(
τ
j
)
i
(17)
for z
i
C, i, j = 1,2,3,4. With the purpose of finding
the number of users present in the network when the
server ends attending queue Q
1
of systems Γ
1
, it is
gotten that
F
1
= E
h
z
L
1
(τ
1
)
1
z
L
2
(τ
1
)
2
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= E
h
z
L
2
(τ
1
)
2
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= E
h
z
L
2
(τ
1
)+X
2
(τ
1
τ
1
)+Y
2
(τ
1
τ
1
)
2
·z
L
3
(τ
1
)+X
3
(τ
1
τ
1
)+Y
3
(τ
1
τ
1
)
3
z
L
4
(τ
1
)+X
4
(τ
1
τ
1
)+Y
4
(τ
1
τ
1
)
4
i
.
(18)
This is obtained using equation (7). Now, for
z
1
,z
2
,z
3
,z
4
C,
F
1
= E
h
z
L
2
(τ
1
)
2
z
X
2
(τ
1
τ
1
)
2
z
Y
2
(τ
1
τ
1
)
2
z
L
3
(τ
1
)
3
z
X
3
(τ
1
τ
1
)
3
z
Y
3
(τ
1
τ
1
)
3
z
L
4
(τ
1
)
4
z
X
4
(τ
1
τ
1
)
4
z
Y
4
(τ
1
τ
1
)
4
i
= E
h
z
L
2
(τ
1
)
2
n
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
on
z
X
2
(τ
1
τ
1
)
2
z
Y
2
(τ
1
τ
1
)
2
on
z
X
3
(τ
1
τ
1
)
3
z
Y
3
(τ
1
τ
1
)
3
o
n
z
X
4
(
τ
1
τ
1
)
4
z
Y
4
(τ
1
τ
1
)
4
oi
= E
h
z
L
2
(τ
1
)
2
n
z
X
2
(τ
1
τ
1
)
2
z
Y
2
(τ
1
τ
1
)
2
o
n
z
X
3
(τ
1
τ
1
)
3
z
Y
3
(τ
1
τ
1
)
3
on
z
X
4
(τ
1
τ
1
)
4
z
Y
4
(τ
1
τ
1
)
4
oi
E
h
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
.
(19)
The last equation was obtained applying the fact that
the arrivals processes of the queues in each of the sys-
tems are assumed to be independent. Hence, it is pos-
sible to separate the expectation for the arrival pro-
cesses at time τ
1
, which is the time the server visits
Q
1
. Recall that
˜
X
i
(t) = X
i
(t) + Y
i
(t) for i = 2,3,4,
then it is obtained for z
1
,z
2
,z
3
,z
4
C that
F
1
= E
h
z
L
2
(τ
1
)
2
n
z
˜
X
2
(τ
1
τ
1
)
2
z
˜
X
3
(τ
1
τ
1
)
3
z
˜
X
4
(τ
1
τ
1
)
4
oi
· E
h
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= E
h
z
L
2
(τ
1
)
2
n
˜
P
2
(z
2
)
τ
1
τ
1
˜
P
3
(z
3
)
τ
1
τ
1
˜
P
4
(z
4
)
τ
1
τ
1
oi
·E
h
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= E
h
z
L
2
(τ
1
)
2
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
τ
1
τ
1
i
·E
h
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= E
h
z
L
2
(τ
1
)
2
˜
θ
1
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
L
1
(τ
1
)
i
· E
h
z
L
3
(τ
1
)
3
z
L
4
(τ
1
)
4
i
= F
1
˜
θ
1
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
,z
2
·F
1
(z
3
,z
4
;τ
1
)
=: F
1
˜
θ
1
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
,z
2
,z
3
,z
4
.
(20)
The last equalities are true because the number of arri-
vals to Q
4
during the time interval [τ
1
,
τ
1
] still have
not been attended by the server in Γ
2
, then the users
cannot transfer to Γ
1
through the queue Q
2
. There-
fore the number of users switching from Q
4
to Q
2
during the time interval [τ
1
,τ
1
] depends on the policy
of transferring between the two systems. The server’s
switchover times are given by the general equations
R
i
(z
1
,z
2
,z
3
,z
4
) = R
i
˜
P
1
(z
1
)
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
,
z
i
C, i = 1, 2, 3, 4. Then, to derive and evaluate in
z
i
= 1, it is obtained that
D
j
R
j
= r
i
˜µ
j
,i, j = 1,2,3,4. (21)
And the second order partial derivatives are given by
D
j
D
i
R
k
= R
(2)
k
˜µ
i
˜µ
j
+ 11
i= j
r
k
P
(2)
i
+ 11
i6= j
r
k
˜µ
i
˜µ
j
, (22)
for any i, j,k = 1, 2, 3, 4, where 11
i= j
= 1 for i = j,
and 0 in any other case. (Observe that in the last
derivatives the evaluation in z
i
= 1, for i,k = 1, 2, 3, 4
is omitted, in order to simplify the notation.) Then the
joint PGF for Q
1
in Γ
1
is given by
F
1
(z
1
,z
2
,z
3
,z
4
) = R
2
4
i=1
˜
P
i
(z
i
)
·F
2
z
1
,
˜
θ
2
˜
P
1
(z
1
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
,z
3
,z
4
,
(23)
for z
i
C, i = 1,2,3,4. For the rest of the queues simi-
lar expressions are gotten by an analogous argument.
Now the switchover times from one queue to the other
are considered, as well as the number of users present
at the time the server leaves the queue to start atten-
ding the next one. In analogous way for the rest of the
Network of M/M/1 Cyclic Polling Systems
301
NCPS it is obtained for z
i
C, i = 1,2,3,4, that
F
2
(z
1
,z
2
,z
3
,z
4
) = R
1
4
i=1
˜
P
i
(z
i
)
·F
1
˜
θ
1
˜
P
2
(z
2
)
˜
P
3
(z
3
)
˜
P
4
(z
4
)
,z
2
,z
3
,z
4
,
F
3
(z
1
,z
2
,z
3
,z
4
) = R
4
4
i=1
˜
P
i
(z
i
)
· F
4
(z
1
,z
2
,
z
3
,
˜
θ
4
˜
P
1
(z
1
)
˜
P
2
(z
2
)
˜
P
3
(z
3
)

,
F
4
(z
1
,z
2
,z
3
,z
4
) = R
3
4
i=1
˜
P
i
(z
i
)
· F
3
(z
1
,z
2
,
˜
θ
3
˜
P
1
(z
1
)
˜
P
2
(z
2
)
˜
P
4
(z
4
)
,z
4
.
(24)
From (16), the following derivatives are obtained:
D
j
F
i
(z
1
,z
2
;τ
i+2
) = 11
j2
F
(1)
j,i+2
and
D
j
F
i
(z
3
,z
4
;τ
i2
) = 11
j3
F
(1)
j,i2
,
(25)
for z
i
C, i = 1,2,3,4, and the second order deriva-
tives are given by
D
j
D
i
F
k
(z
1
,z
2
;τ
k+2
) = 11
k2
11
j2
11
i2
11
j=i
F
(2)
i,k+2
+11
j6=i
F
(1)
j,k+2
F
(1)
i,k+2
,
D
j
D
i
F
k
(z
3
,z
4
;τ
k2
) = 11
k3
11
j3
11
i3
11
j=i
F
(2)
i,k2
+11
j6=i
F
(1)
j,k2
F
(1)
i,k2
,
(26)
for z
i
C, i = 1, 2, 3, 4. The following theorem shows
how to find the lengths of the queues of the NCPS at
polling instants according to equations (23) and (24):
Theorem 1. Suppose that ˜µ = ˜µ
1
+ ˜µ
2
< 1, ˆµ =
˜µ
3
+ ˜µ
4
< 1, then the number of users in the queues
conforming the NCPS (24) at polling instants is
f
j
(i) = r
j+1
˜µ
i
+ 11
i6= j+1
f
j+1
( j + 1)
˜µ
i
1˜µ
j+1
+11
i= j
f
j+1
(i) + 11
j=1
11
i3
F
(1)
i, j+1
+11
j=3
11
i2
F
(1)
i, j+1
(27)
for j = 1,3 and i = 1,2,3,4, and
f
j
(i) = r
j1
˜µ
i
+11
i6= j1
f
j1
( j 1)
˜µ
i
1˜µ
j1
+11
i= j
f
j1
(i) + 11
j=2
11
i3
F
(1)
i, j1
+11
j=4
11
i2
F
(1)
i, j1
(28)
for j = 2, 4 and i = 1,2,3,4. The solution of the linear
system of equations (27) and (28) is given by:
f
i
( j) = (11
j=i1
+ 11
j=i+1
)r
j
˜µ
j
+11
i= j
11
i2
r˜µ
i
(1˜µ
i
)
1˜µ
+ 11
i2
ˆr˜µ
i
(1˜µ
i
)
1ˆµ
+11
i=1
11
j3
˜µ
j
r
i+1
+
r˜µ
i+1
1˜µ
+ F
(1)
j,i+1
+11
i=3
11
j3
˜µ
j
r
i+1
+
ˆr˜µ
i+1
1ˆµ
+ F
(1)
j,i+1
+11
i=2
11
j2
˜µ
j
r
i1
+
r˜µ
i1
1˜µ
+ F
(1)
j,i1
+11
i=4
11
j2
˜µ
j
r
i1
+
ˆr˜µ
i1
1ˆµ
+ F
(1)
j,i1
,
for i, j = 1,2,3,4.
Theorem 2. Suppose ˜µ, ˆµ < 1, then from the ex-
pressions given in (23) and (24) the second order
derivatives for the NCPS are obtained which, in their
general form, are
f
1
(i,k) = D
k
D
i
(R
2
+ F
2
+ 11
i3
F
4
)
+D
i
R
2
D
k
(F
2
+ 11
k3
F
4
)
+D
i
F
2
D
k
(R
2
+ 11
k3
F
4
)
+11
i3
D
i
F
4
D
k
(R
2
+ F
2
),
f
2
(i,k) = D
k
D
i
(R
1
+ F
1
+ 11
i3
F
3
)
+D
i
R
1
D
k
(F
1
+ 11
k3
F
3
)
+D
i
F
1
D
k
(R
1
+ 11
k3
F
3
)
+11
i3
D
i
F
3
D
k
(R
1
+ F
1
),
f
3
(i,k) = D
k
D
i
(R
4
+ 11
i2
F
2
+ F
4
)
+D
i
R
4
D
k
(11
k2
F
2
+ F
4
)
+D
i
F
4
D
k
(R
4
+ 11
k2
F
2
)
+11
i2
D
i
F
2
D
k
(R
4
+ F
4
),
f
4
(i,k) = D
k
D
i
(R
3
+ 11
i2
F
1
+ F
3
)
+D
i
R
3
D
k
(11
k2
F
1
+ F
3
)
+D
i
F
3
D
k
(R
3
+ 11
k2
F
1
)
+11
i2
D
i
F
1
D
k
(R
3
+ F
3
),
(29)
for i,k = 1, 2, 3, 4. The second order moments are ob-
tained solving the linear systems given by (29).
The proof is given in Appendix A.
4 EXPECTED QUEUE LENGTHS
AT ANY TIME
Assumption 1. (i) The arrival processes in the
NCPS satisfies ˜µ, ˆµ < 1.
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
302
(ii) Each of the queues of the NCPS is an M/M/1 sys-
tem, with
˜
ρ
i
:= ˜µ
i
/λ
i
< 1, for i = 1,2,3,4 (ob-
serve that in the case considered
˜
ρ
i
= ˜µ
i
, for
i = 1,2,3,4, given that the service time is assumed
to be proportional to the length of the slot).
(iii) The switchover times have a finite first moment.
In this section it is supposed that Assumption 1
holds. Here, the idea given in (Takagi, 1986) is fo-
llowed, in order to find the expected queue lengths at
any time for the NCPS.
Fix i
{
1,2,3,4
}
. Let L
i
be the number of users
at queue Q
i
at polling instants, then, following Section
3, it is obtained that
E [L
i
] = f
i
(i),
Var [L
i
] = f
i
(i,i) + E [L
i
] E [L
i
]
2
.
(30)
Consider the cycle time C
i
for queue Q
i
with dura-
tion given by τ
i
(m + 1) τ
i
(m) for m 1. The inter-
val between two successive regeneration points will
be called regenerative cycle.
In order to guarantee that the cyclic times are a sta-
tionary state, it is assumed that each of the polling sys-
tems are stable (Boon et al., 2011; Boxma et al., 1992;
Cooper et al., 1996; Levy and Sidi, 1990), which en-
sure the stability of each queue (Fricker and Jaibi,
1994; Vishnevskii and Semenova, 2006), and this im-
plies the stationarity of the cyclic times (Altman et al.,
1992). Therefore it is assumed that the cyclic times
are stationary as Takagi did (Takagi, 1986).
Let M
i
be the number of polling cycles in a regen-
erative cycle. The duration of the m-th polling cycle
in a regeneration cycle will be denoted by C
(m)
i
, for
m = 1,2,.. . , M
i
. The mean polling cycle time is de-
fined by
E [C
i
] =
M
i
m=1
E
h
C
(m)
i
i
E [M
i
]
. (31)
For the process L
i
(t), t 0, their PGF will be de-
noted by Q
i
(z), z C, which is also given by the time
average of z
L
i
(t)
over the regenerative cycle defined
before, so it is obtained that
Q
i
(z) = E
h
z
L
i
(t)
i
=
E
h
M
i
m=1
τ
i
(m+1)1
t=τ
i
(m)
z
L
i
(t)
i
E
h
M
i
m=1
(τ
i
(m+1)τ
i
(m))
i
,
(32)
which can be rewritten in the form
Q
i
(z) =
1
E [C
i
]
·
1 F
i
(z)
P
i
(z) z
·
(1 z) P
i
(z)
1 P
i
(z)
, (33)
(see Section 3 in (Takagi, 1986)). The following
proposition provides the expected queue lengths for
each of the queues in the NCPS at any time.
Theorem 3. For the queue lengths in the NCPS at
any time, with PGF given in (33), the first and second
order moments are given by
Q
(1)
i
(1) =
1
˜µ
i
(1˜µ
i
)
E(L
i
)
2
2E[C
i
]
σ
2
i
E
[
L
i
]
2E[C
i
]
·
12˜µ
i
(1˜µ
i
)
2
˜µ
2
i
,
(34)
where σ
2
i
=
Var
˜
X
i
(t)

2
, and
E [C
i
]Q
(2)
i
(1) =
1
˜µ
3
i
(1˜µ
i
)
3
n
(1 ˜µ
i
)
2
˜µ
2
i
O
(2)
1,i
(1)
˜µ
i
(1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(2)
3,i
(1)
˜µ
2
i
(1 ˜µ
i
)
2
O
1,i
(1)P
(2)
i
(1)
+ 2˜µ
i
(1 2˜µ
i
)O
1,i
(1) (1 ˜µ
i
)
O
(1)
3,i
(1)
2
2 (1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(1)
3,i
(1)
2˜µ
3
i
(1 ˜µ
i
)
2
O
(1)
1,i
(1)
2 (1 2˜µ
i
)O
(1)
3,i
(1)O
(1)
1,i
(1)
2˜µ
2
i
(1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(1)
1,i
(1)
o
,
(35)
for i = 1,2,3,4.
Proof. Fix i
{
1,2,3,4
}
and z C. To remove the
singularities in (33) it is necessary to define the follo-
wing analytic functions:
ϕ
i
(z) = 1 F
i
(z), ψ
i
(z) = z P
i
(z),
and ς
i
(z) = 1 P
i
(z),
(36)
then
E [C
i
]Q
i
(z) =
(z 1) ϕ
i
(z)P
i
(z)
ψ
i
(z)ς
i
(z)
. (37)
For k 0, define a
k
= P
{
L
i
(t) = k
}
. It is obtained
that
ϕ
i
(z) = 1 F
i
(z) = 1
+
k=0
a
k
z
k
,
therefore
ϕ
(1)
i
(z) =
+
k=1
ka
k
z
k1
,with
ϕ
(1)
i
(1) = E [L
i
(t)], and
ϕ
(2)
i
(z) =
+
k=2
k(k 1)a
k
z
k2
, hence
ϕ
(2)
i
(1) = E [L
i
(L
i
1)].
In the same way it is gotten that
ϕ
(3)
i
(z) =
+
k=3
k(k 1)(k 2)a
k
z
k3
and
ϕ
(3)
i
(1) = E [L
i
(L
i
1)(L
i
2)].
Network of M/M/1 Cyclic Polling Systems
303
Expanding ϕ
i
(z) around z = 1,
ϕ
i
(z) = ϕ
i
(1) +
ϕ
(1)
i
(1)
1!
(z 1)
+
ϕ
(2)
i
(1)
2!
(z 1)
2
+
ϕ
(3)
(1)
3!
(z 1)
3
+ ... +
= (z 1)
(
ϕ
(1)
i
(1) +
ϕ
(2)
(1)
2!
(z 1)
+
ϕ
(3)
i
(1)
3!
(z 1)
2
+ ...+
)
= (z 1)O
1,i
(z)
with O
1,i
(z) 6= 0, given that O
1,i
(z) = E [L
i
], where
O
1,i
(z) = ϕ
(1)
i
(1) +
ϕ
(2)
i
(1)
2!
(z 1)
+
ϕ
(3)
i
(1)
3!
(z 1)
2
+ ... + .
(38)
Calculating the derivatives of O
1,i
(z), and evaluating
in z = 1, it is obtained that
O
1,i
(1) = E [L
i
],
O
(1)
1,i
(1) =
1
2
E
(L
i
)
2
+
1
2
E [L
i
]
and
O
(2)
1,i
(1) =
1
3
E
(L
i
)
3
+ E
(L
i
)
2
2
3
E [L
i
].
(39)
Proceeding in a similar manner for ψ
i
(z) = z P
i
(z)
and ς
i
(z) = 1 P
i
(z), it is gotten that
E [C
i
]Q
i
(z) =
O
1,i
(z)P
i
(z)
O
2,i
(z)O
3,i
(z)
. (40)
Calculating the derivative with respect to z, and eva-
luating in z = 1,
E [C
i
]Q
(1)
i
(1) =
1
(1˜µ
i
)
2
˜µ
2
i

1
2
E
(L
i
)
2
+
1
2
E [L
i
]
(1 ˜µ
i
)(˜µ
i
)(E [L
i
])(1 ˜µ
i
)(˜µ
i
) ˜µ
i
1
2
E
˜
X
2
i
(t)
+
1
2
˜µ
i
(˜µ
i
)(E [L
i
])
(1 ˜µ
i
)(E [L
i
])
1
2
E
˜
X
2
i
(t)
+
1
2
˜µ
i

=
1
(1˜µ
i
)
2
˜µ
2
i
1
2
˜µ
2
i
E
(L
i
)
2
+
1
2
˜µ
i
E
(L
i
)
2
+
1
2
˜µ
2
i
E [L
i
] ˜µ
3
i
E [L
i
]
+ ˜µ
i
E [L
i
]E
˜
X
2
i
(t)
1
2
E [L
i
]E
˜
X
2
i
(t)

=
1
2˜µ
i
(1˜µ
i
)
E
(L
i
)
2
1
2
˜µ
i
(1˜µ
i
)
2
˜µ
2
i
σ
2
i
E [L
i
].
It means that
Q
(1)
i
(1) =
1
˜µ
i
(1 ˜µ
i
)
E
(L
i
)
2
2E [C
i
]
σ
2
i
EL
i
2E [C
i
]
·
1 2˜µ
i
(1 ˜µ
i
)
2
˜µ
2
i
.
Deriving again and evaluating in z = 1, it follows that
E [C
i
]Q
(2)
i
(1) =
1
˜µ
3
i
(1˜µ
i
)
3
n
(1 ˜µ
i
)
2
˜µ
2
i
O
(2)
1,i
(1)
˜µ
i
(1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(2)
3,i
(1)
˜µ
2
i
(1 ˜µ
i
)
2
O
1,i
(1)P
(2)
i
(1)
+ 2˜µ
i
[(1 2˜µ
i
)O
1,i
(1) (1 ˜µ
i
)]
O
(1)
3,i
(1)
2
2 (1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(1)
3,i
(1)
2˜µ
3
i
(1 ˜µ
i
)
2
O
(1)
1,i
(1)
2 (1 2˜µ
i
)O
(1)
3,i
(1)O
(1)
1,i
(1)
2˜µ
2
i
(1 ˜µ
i
)(1 2˜µ
i
)O
1,i
(1)O
(1)
1,i
(1)
o
,
where O
1,i
(1),O
(1)
1,i
(1),O
(1)
3,i
(1),O
(2)
3,i
(1),P
(2)
i
(1) can
be obtained using direct operations.
Remark 1. To determine the second order mo-
ments for the queue lengths, it is necessary to cal-
culate the third derivative of the arrival processes for
each of the queues.
5 CONCLUDING REMARKS
This proposal about polling systems, that could be
adressed to polling stations, using the buffer occu-
pancy method allow to find analytical expressions for
the first and second moments of the queue lengths at
any time t > 0. The extension of these results to other
policies and the continuous case are object of future
work.
ACKNOWLEDGEMENTS
Research supported by the UACM/ICyT/SECITI-D.F.
through the project PI2014-1
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APPENDIX A: GENERAL CASE
CALCULATIONS FOR THE PGF
Recall that (25) and (29) give the first and the se-
cond order partial derivatives, respectively. The first
moments equations for the expected queue lengths at
polling instants are obtained solving the system given
in Theorem 1. The second moment for queue Q
1
at
polling instants is given by
f
1
(1,1) =
˜µ
1
1 ˜µ
2
2
f
2
(2,2) + 2
˜µ
1
1 ˜µ
2
f
2
(2,1)
+ f
2
(1,1) + ˜µ
2
1
R
(2)
2
+ f
2
(2)θ
(2)
2
+
˜
P
(2)
1
f
2
(2)
1 ˜µ
2
+ r
2
+ 2r
2
˜µ
2
f
2
(1).
Similar argument allows to obtain the following
general expressions for Q
1
:
f
1
(i, j) = 11
i=1
f
2
(1,1)
+
h
(1 11
i= j=3
)11
i+ j6
11
i j
µ
j
1˜µ
2
+(1 11
i= j=3
)11
i+ j6
11
i> j
µ
i
1˜µ
2
+11
i=1
µ
i
1˜µ
2
i
f
2
(1,2) + 11
i, j6=2
1
1˜µ
2
2
µ
i
µ
j
f
2
(2,2)
+
h
11
i, j6=2
˜
θ
(2)
2
˜µ
i
˜µ
j
+11
i, j6=2
11
i= j
˜
P
(2)
i
1˜µ
2
+11
i, j6=2
11
i6= j
˜µ
i
˜µ
j
1˜µ
2
i
f
2
(2) +
h
r
2
˜µ
i
+ 11
i3
F
(1)
i,2
i
f
2
( j)
+
h
r
2
˜µ
j
+ 11
j3
F
(1)
j,2
i
f
2
(i) +
h
R
(2)
2
+ 11
i= j
r
2
i
˜µ
i
µ
j
+11
j3
F
(1)
j,2
h
11
j6=i
F
(1)
i,2
+ r
2
˜µ
i
i
+r
2
h
11
i= j
P
(2)
i
+ 11
i3
F
(1)
i,2
˜µ
j
i
+ 11
i3
11
j=i
F
(2)
i,2
.
(41)
In a similar manner, expressions for f
2
(i, j), f
3
(i, j)
and f
4
(i, j) are obtained for i, j = 1, 2, 3,4. These expres-
sions give place to a linear system of equations whose
some of the solutions are
f
1
(1,1) = b
3
, f
2
(2,2) = η
1
,
f
3
(3,3) = η
2
, f
4
(4,4) = a
38
η
2
+ a
39
K
29
,
where
η
1
=
b
2
1b
1
, η
2
=
b
5
1b
4
,
N
1
= a
2
K
12
+ a
3
K
11
+ K
1
, N
2
= a
12
K
2
+ a
13
K
5
+ K
15
,
b
1
= a
1
a
11
, b
2
= a
11
N
1
+ N
2
,
b
3
= a
1
b
2
1b
1
+ N
1
, N
3
= a
29
K
39
+ a
30
K
38
+ K
28
N
4
= a
39
K
29
+ a
40
K
30
+ K
40
, b
4
= a
28
a
38
,
b
5
= a
28
N
4
+ N
3
, b
6
= a
38
b
5
1b
4
+ N
4
.
Network of M/M/1 Cyclic Polling Systems
305