Centrum voor Wiskunde
en
lnformatica
Centre
for
Mathematics
and
Computer
Science
J.L. van den Berg, O.J. Boxma
Sojourn times
in
feedback queues
Department of Operations Research and
S~m
Theory
Report
OS-R871
O
June
The Centre for Mathematics
and
Computer Science
is
a research institute of the Stichting
Mathematisch Centrum, which
was
founded
on
February
11
, 1946,
as
a nonprofit institution aim-
ing
at
the promotion of mathematics, computer science,
and
their applications. It
is
sponsored by
the Dutch Government through
the·
Netherlands Organization for the Advancement of Pure
Research (Z.W.0.).
Copyrigpt
© Stichting Mathematisch Centrum, Amsterdam
Sojourn Times in Feedback Queues
J.L. van den Berg,
0.J.
Boxma
Centre for Mathematics
and
Computer Science
P.O.
Box 4079, 1009
AB
Amsterdam, The Netherlands
This paper considers
an
M/M/1
queue with a very general feedback mechanism. When a customer com-
pletes his i-th service, he departs from the system with probability 1 -
p(1) and he cycles back with proba-
bility
p(1). The main result of the paper
is
a formula for the joint distribution of the successive sojourn
times of a customer in the system. As a by-result, it is shown that the sojourn times in all individual cycles
are identically, negative exponentially, distributed. Also, the correlation between the sojourn times of the
j-
th and k-th cycle of a customer is calculated; furthermore, the distribution of the total sojourn time
is
derived.
1980 Mathematics Subject Classification:
601<25,
68M20.
Key Words & Phrases:
M/M/1
queue, feedback, sojourn times.
Note: This report will be submitted for publication elsewhere.
1.
INTRODUCTION
1
This paper considers an
MIMI
1 queue with a very general feedback mechanism. When a newly arriv-
ing customer, to be called a type-I customer,
has
received his service, he departs from the system with
probability
1-p(l)
and is fed back to the end of the queue with probability
p(l);
in the latter case he
becomes a type-2 customer. When he has received his
i-th service, he leaves with probability 1
-p
(i)
and he cycles back with probability p (i), in the latter case becoming a type-i + 1 customer. The ser-
vice times of each customer at all visits are identically, negative exponentially, distributed stochastic
variables. The resulting queueing model has the property that the joint queue-length distribution of
type-i customers,
i = 1,2, ... , is of product-form type. This property is exploited to analyse the
sojourn-time process.
In
particular,
we
are interested
in
the joint distribution
of
the sojourn times of
a customer on his successive cycles.
Feedback queues are useful for modelling many phenomena in computer-communication systems.
The following example in computer timesharing was communicated to us by R.D. Nelson (IBM
Research Center, Hawthorne). A newly arriving
job
is considered to be a class 1 job. The scheduler
first allocates some CPU time to the job. During
this
initial CPU allocation the job,
if
not completed,
establishes its initial working set and typically changes class to class
2.
The next time the job obtains
the processor
it
is allocated a CPU slice for class 2 customers and,
if
not finished, becomes a class 3
job with its associated CPU slice.
This
procedure continues until the job leaves the system. This
example
gives
rise to a single server queue with feedback. R.D. Nelson [7] considers both the case in
which jobs of all classes form one queue which is processed in FCFS order, and the case of a head-
of-the-line priority scheme. For both cases, allowing general service time distributions, he has
obtained a set of equations for the mean sojourn times during each cycle. This set of equations can be
solved numerically.
Other examples
of
feedback systems are found in telecommunications. E.g., a telephone call may
generate several tasks for processing. Such tasks can sometimes be considered as feedbacks. From a
customer's viewpoint, a very important performance measure is the response time or sojourn time,
defined as the time spent by a particular sequence of tasks from its arrival
to its service completion
[6];
and not just its mean is of interest, but also (the tail of) its distribution.
In
the queueing literature, research on feedback queues has been mainly restricted to single-server
queues with so-called Bernoulli feedback [3,4,9]: when a customer (task) completes his service, he
departs from the system with probability
1-p
and is fed back with probability p. Fontana and Diaz
Report 08-R871 O
Centre for Mathematics and Computer Science
P.O.
Box 4079, 1009
AB
Amsterdam, The Netherlands
2
Berzosa
[5,6]
extend some results obtained for·the
MIG/l
model with Bernoulli feedback to a more
general feedback model with priorities. In
[2]
we
have considered sojourn time distributions in an
MIMI
1
queue with deterministic feedback,
viz.,
each customer makes exactly N passes through the
system. Simultaneously, Doshi and Kaufman
[4]
have studied sojourn time distributions in
an
MIG!l
queue with Bernoulli feedback, employing an iteration procedure which
is
very similar to the
one used in
[2].
The feedback mechanisms in those studies are special cases of the one in the present
study: for deterministic feedback, take
p(i)=
1,
i~N
-1,
p(N)=O;
for Bernoulli feedback, take
p(i)=P.
The organization of the rest of the paper is as follows. In Section 2 the model is described in
detail. Section 3 contains our main result.
We
derive a formula for (the transform of) the joint distri-
bution of the successive sojourn times of a customer in the system and the number of customers of
the various types present at his successive departure epochs. As a by-result, it
is
shown that the
sojourn times in all individual cycles are identically, negative exponentially, distributed. Also, the
correlation between the sojourn times of the
j-th
and k-th cycle of a customer
is
calculated; further-
more, the distribution of the total sojourn time is derived. In Section 4 two special feedback mechan-
isms are studied: Bernoulli feedback (Subsection
4.1)
and deterministic feedback (Subsection
4.2).
In
these two cases, a limiting procedure leads to the sojourn time distribution in the
MIMI
1
queue with
processor sharing and the
MIDI
1 queue with processor sharing, respectively.
2.
MODEL
DESCRIPTION
We consider a single server queueing system with infinite waiting room,
see
Fig.
1.
Customers arrive
at
the system according to a Poisson process with intensity .;\.>0. After having received a service, a
customer may either leave the system or be fed back. When a customer has completed his i-th ser-
vice,
he departs from the system with probability
l -
p(i)
and is
fed
back with probability
p(i).
Fed
back customers return instantaneously, joining the end of the queue. A customer who
is
visiting the
queue for the i-th time
will
be called a type-i customer. To avoid the problems that occur in dealing
with an infinite number of customer types, it is assumed that after a certain number of services the
feedback probabilities of a customer remain constant. Thus
p(i)
=
p(N)
:
=
p, i
=N,N
+
1,
...
for
some
N~
1.
The service discipline
is
First Come First Served (FCFS).
p(i)
I · · · I
1-p(i)
Fig. 1
MIMI
1
queue with feedback.
The successive service times of a customer are independent, negative exponentially distributed, ran-
dom variables, with mean
{J.
These service times are also independent of the service times of other
customers.
Introduce
q(O)
:=
1,
(2.1)
i-1
q(i):=
Ilp(i),
i=l,
...
,N-1,
j=O
3
co
m-1
q(N)
:
=
~
II
p(j)
=
q(N
-
l)p(N
-1)
I
(1-p),
m=N
j=O
with
p(O)
:=
I.
Note that
q(i) is the relative arrival rate of
type-i
customers,
i
=
l,
...
,N. The offered load to the queue
N
per unit of time is
p:
=A.{3~q(i).
For
stability it is required that
p<
I.
i
==I
We are interested in the following quantities:
-
~:
number
of
type-i
customers in the system
at
an
arbitrary epoch,
i
= l, ... ,N;
-
xy>:
number of
type-i
customers in the system
at
the
j-th
service completion
of
a customer,
i
=
I,
...
,N,
j
=
1,2,
...
;
-
x~
0
>:
number of
type-i
customers in the system
at
the arrival of a new customer,
i
=
l,
...
,N;
- Sj: time required for the
j-th
pass through the system (j-th sojourn time),
j=
1,2,
....
3.
MAIN
RESULTS
In
this section
we
present, in the form of Laplace-Stieltjes transforms and generating functions,
an
expression for the
j~int
distribution of the successive sojourn times
Sj,
j =
l,
... ,k, and the number of
type-i
customers,
xp>,
i
=
l,
...
,N, present
at
the
j-th
service completion of a customer who is fed back
at
least
k-1
times, k
=
1,2,
....
First note that the system described above can be considered as a
queueing network consisting of one queue with
N
types of customers. Type-i customers are fed back
with probability
p(i)
after service, and then change into type-(i
+
1)
customers,
i
=
l,
...
,N
-1.
Type-
N
customers are fed back with probability
p
after service, and do not change their type. Because the
service times are assumed to be exponentially distributed, the results obtained by Baskett et al. [l] can
be applied to find the joint distribution of the number
of
type-i customers in the system
at
an
arbi-
trary epoch.
It
is found that, for
XJ.
•••
,xN
=
0,
1,
...
,
P(xi.
...
,xN)
:=
Pr{X1
=xi.
...
,XN=xN}
=
(3.1)
(1-A.Pf
q(i)) IT(A.{3q(i)t'
(x17
... +xN;!.
i=l
i=I
Xt
••
• •
XN.
The PASTA property
([10])
implies the equality of the joint queue length distribution
at
the epoch of
a new arrival and
at
an
arbitrary epoch:
Pr{X~
0
>=xi.
...
,XW>=xN}
=
P(xi.
...
,xN),
xi.
...
,xN=0,1,....
(3.2)
We follow a customer from the moment he arrives as a type-I customer until he completes his k-th
service.
It
holds that
(3.3)