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

MartÃ nez-RodrÃ 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

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

,

ˆ

P

i

(z

i

) = E

h

z

Y

i

(t)

i

, (1)

and

˜

P

i

(z

i

) = E

h

z

˜

X

i

(t)

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

, 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

, (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

n−1, j

P



˜

X

n

= k − j + 1



=

∑

k+1

j=1

g

n−1, j

P

{

X

n

+Y

n

= k − j + 1

}

=

∑

k+1

j=1

∑

j

l=1

g

n−1, 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

n−1

(z) − G

n−1

(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

, (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

j≤2

F

(1)

j,i+2

and

D

j

F

i

(z

3

,z

4

;τ

i−2

) = 11

j≥3

F

(1)

j,i−2

,

(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

k≤2

11

j≤2

11

i≤2



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

k−2

) = 11

k≥3

11

j≥3

11

i≥3



11

j=i

F

(2)

i,k−2

+11

j6=i

F

(1)

j,k−2

F

(1)

i,k−2



,

(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

i≥3

F

(1)

i, j+1

+11

j=3

11

i≤2

F

(1)

i, j+1

(27)

for j = 1,3 and i = 1,2,3,4, and

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=2

11

i≥3

F

(1)

i, j−1

+11

j=4

11

i≤2

F

(1)

i, j−1

(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=i−1

+ 11

j=i+1

)r

j

˜µ

j

+11

i= j



11

i≤2

r˜µ

i

(1−˜µ

i

)

1−˜µ

+ 11

i≥2

ˆr˜µ

i

(1−˜µ

i

)

1−ˆµ



+11

i=1

11

j≥3



˜µ

j



r

i+1

+

r˜µ

i+1

1−˜µ



+ F

(1)

j,i+1



+11

i=3

11

j≥3



˜µ

j



r

i+1

+

ˆr˜µ

i+1

1−ˆµ



+ F

(1)

j,i+1



+11

i=2

11

j≤2



˜µ

j



r

i−1

+

r˜µ

i−1

1−˜µ



+ F

(1)

j,i−1



+11

i=4

11

j≤2



˜µ

j



r

i−1

+

ˆr˜µ

i−1

1−ˆµ



+ F

(1)

j,i−1



,

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

i≥3

F

4

)

+D

i

R

2

D

k

(F

2

+ 11

k≥3

F

4

)

+D

i

F

2

D

k

(R

2

+ 11

k≥3

F

4

)

+11

i≥3

D

i

F

4

D

k

(R

2

+ F

2

),

f

2

(i,k) = D

k

D

i

(R

1

+ F

1

+ 11

i≥3

F

3

)

+D

i

R

1

D

k

(F

1

+ 11

k≥3

F

3

)

+D

i

F

1

D

k

(R

1

+ 11

k≥3

F

3

)

+11

i≥3

D

i

F

3

D

k

(R

1

+ F

1

),

f

3

(i,k) = D

k

D

i

(R

4

+ 11

i≤2

F

2

+ F

4

)

+D

i

R

4

D

k

(11

k≤2

F

2

+ F

4

)

+D

i

F

4

D

k

(R

4

+ 11

k≤2

F

2

)

+11

i≤2

D

i

F

2

D

k

(R

4

+ F

4

),

f

4

(i,k) = D

k

D

i

(R

3

+ 11

i≤2

F

1

+ F

3

)

+D

i

R

3

D

k

(11

k≤2

F

1

+ F

3

)

+D

i

F

3

D

k

(R

3

+ 11

k≤2

F

1

)

+11

i≤2

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

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

]

·

1−2˜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

k−1

,with

ϕ

(1)

i

(1) = −E [L

∗

i

(t)], and

ϕ

(2)

i

(z) = −

∑

+∞

k=2

k(k − 1)a

k

z

k−2

, 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

k−3

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

REFERENCES

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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+ j≤6

11

i≤ j

µ

j

1−˜µ

2

+(1 − 11

i= j=3

)11

i+ j≤6

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

i≥3

F

(1)

i,2

i

f

2

( j)

+

h

r

2

˜µ

j

+ 11

j≥3

F

(1)

j,2

i

f

2

(i) +

h

R

(2)

2

+ 11

i= j

r

2

i

˜µ

i

µ

j

+11

j≥3

F

(1)

j,2

h

11

j6=i

F

(1)

i,2

+ r

2

˜µ

i

+r

2

h

11

i= j

P

(2)

i

+ 11

i≥3

F

(1)

i,2

˜µ

j

i

+ 11

i≥3

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

1−b

1

, η

2

=

b

5

1−b

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

1−b

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

1−b

4



+ N

4

.

Network of M/M/1 Cyclic Polling Systems

305