Abstract-In this letter, a new recursive tracking algorithm is presented that is capable of estimating the real arrival rate, real , to the system. The estimated value of the arrival rate, estimated , is used to dynamically adjust the control parameters of the system, hence ensuring that the operating point of the system is pushed toward the required settling point, whatever the real arrival rate to the system. This algorithm utilizes system information through the feedback channel in order to dynamically adjust the estimated value of the arrival rate and hence update the values of the control parameters

Member, IEEE Atheer Al-Amir

Member, IEEE Mustafa K Gurcan

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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001 387
Arrival Rate Estimation Algorithm for Single Group
Slotted ALOHA Systems
Atheer Al-Amir, Member, IEEE,and Mustafa K. Gurcan, Member, IEEE
Abstract—In this letter, a new recursive tracking algorithm is
presented that is capable of estimating the real arrival rate, real,
to the system. The estimated value of the arrival rate, estimated,
is used to dynamically adjust the control parameters of the system,
hence ensuring that the operating point of the system is pushed
toward the required settling point, whatever the real arrival rate
to the system. This algorithm utilizes system information through
the feedback channel in order to dynamically adjust the estimated
value of the arrival rate and hence update the values of the control
parameters.
Index Terms—Multiple access, multiplexing, stabilization of
Aloha, slotted Aloha.
I. INTRODUCTION
REFERENCE [3] identifies how a slotted Aloha system canbe stabilized for a given arrival rate. Stability of the system
is achieved by dynamically adjusting control parameters to de-
termine the retransmission probability. These retransmission
probability calculations depend on the estimated number,,
of backlogged users which in turn is adjusted using control pa-
rameters. These control parameters (, and relate, re-
spectively, to the no-transmission, one successful transmission
and the collision situation during any time slot) are chosen for
a given value of the arrival rate [1]–[3]. If a system is operated
with an estimated arrival rate value different to the real value, the
system will not necessarily settle at the optimum offered load,
that maximizes the throughput of the system. The max-
imum throughput of a slotted Aloha system is equal to 0.368
packets/slot and is achieved when .
In this letter, it is assumed that a large but finite number of
users transmit their packets of identical length utilizing the ideal
slotted Aloha system, operating in Deferred First Transition [2].
The channel propagation delay is assumed to be negligible. Each
user starts packet transmission at the start of a time slot and
ends before the start of the next time slot. Each user has at most
one packet to transmit at any time. The number of new packets
arriving to the backlog state over a single time slot is taken to
be Poisson distributed with a mean rate of packets/slot.
A collision occurs on the channel if more than one packet is
transmitted in the same time slot, by different users, otherwise
the packet is assumed to have been received correctly in that slot.
Users whose packets have suffered a collision will retransmit
Manuscript received April 16, 2001. The associate editor coordinating the
review of this paper and approving it for publication was Dr. J. Evans.
The authors are with the Communications and Signal Processing Research
Group, Department of Electrical and Electronic Engineering, Imperial College
of Science, Technology and Medicine, London SW7 2BT, U.K. (e-mail:
m.gurcan@ic.ac.uk).
Publisher Item Identifier S 1089-7798(01)09052-4.
their packets at a later time with some probability known as the
retransmission probability,. The joint drift equation is denoted
by and its roots identify the offered loads at which the system
settles [3].
This letter presents a new arrival rate tracking algorithm that
is capable of estimating the arrival rate to the system continu-
ously, using the latest updated estimate to select new values of
the control parameters.
II. REGIONS OFTRACKING ALGORITHM
An assumed initial value of the arrival rate, , is taken to
be between the minimum, packets/slot and maximum,
packets/slot, values of Using the initial
arrival rate, , in the control parameter selection algorithm
[3], initial control parameters values are obtained using
(1)
(2)
For these control parameters, the system will settle at a point
between the lower, and upper, , bounds of the
offered load. These bounds correspond to the roots of the joint
drift equation [3], taking into consideration ,
(3)
when the system is assumed to be operating at and
respectively. In (3), is the offered load which can
take values between zero and infinity and is the drift of the
real number of backlogged users with this being expressed as
[4]
(4)
The drift, , of the estimated number of users is [4]
(5)
These settling points are shown in Fig. 1. From this figure,
two lemmas can be deduced which are of great significance to
the development of the tracking algorithm.
Lemma 1: If the Aloha system operates at an offered load
value, , using control parameters selected utilizing
, then the real arrival rate, , to the system must
satisfy
for (6)
Proof: Let
and . Also let
and . For the control
1089–7798/01$10.00 © 2001 IEEE
388 IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 9, SEPTEMBER 2001
Fig. 1. Drift curves displaying the corresponding roots of the joint drift
equation for minimum and maximum arrival rates to the system and the three
general regions of the tracking algorithm.
parameters selected using , the joint drift equation
satisfies
at (7)
When , the value of the drift, of the real
number, , of backlogged users will satisfy the relationship
at (8)
From (3), (5), and (8), the following relationship for can
be established:
at (9)
since and
at (10)
Equations (9) and (10) imply that the root of lies between
and when .
Lemma 2: If a system operates at an offered load value,
, using control parameters selected utilising ,
then the real arrival rate, , to the system must satisfy
for
(11)
Proof: Can be proven in a similar way to Lemma 1.
The proposed tracking algorithm will perform different func-
tions in different regions depending on the load offered to the
system, as well as the number of users in the backlog. In the
proposed arrival rate tracking algorithm, three regions of oper-
ation exist, as shown in Fig. 1. Transition from regions 1 to 2
is determined by the value of , whereas the transition from
regions 2 to 3 is determined by an arbitrary value ofthat dif-
ferentiates between the backlogged users being in a high or low
population region and is denoted as .
Region 1: Region 1 corresponds to when
and (when system is initiated in the heavy
traffic region, which is assumed in this paper). Users in the
system will measure the offered load of the system using feed-
back information broadcast from the base station. Once
is reached, region 1 is terminated and region 2 is initiated.
Region 2: Region 2 is defined by and
when . In region 2, the
algorithm will update the arrival rate recursively according to
the offered load at which the system is operating. Region 2
will be terminated when either or reaches
.
Region 3: The system is in region 3 if
and . In this regions the joint drift equation
will satisfy . In region 3, the estimated number, ,
of backlogged users will tend toward the minimum allowable
value, . Therefore the retransmission probability will con-
verge toward unity ( ). This implies that will be
equal to zero. Since is also equal to zero in this region,
must be equal to zero in order to satisfy the joint drift equation
. The fact that implies that the
arrival rate is equal to the departure rate. Region 3 is terminated
if increases beyond .
III. STEPS OFPROPOSEDALGORITHM
The steps of the proposed tracking algorithm are as:
Step 1: Initialize the system with initial values of the con-
trol parameters in accordance with (1) and (2) for
. Estimate the offered load of
the system using the maximum likelihood estima-
tion method [5].
Step 2: Once the system starts to operate in the region spec-
ified by , the system moves
to region 2 of operation (if ).
Step 3: In region 2, an initial estimate of the arrival rate
to the system is carried out using the joint drift
equation, [3], from
(12)
Step 4: Find the error signal indicator,, based on Lemmas
1 and 2, from
for
for
(13)
Step 5: Update using
(14)
where represents a constant scaling factor of
the correction applied.
Step 6: Calculate a new value of (keeping and
fixed at the initial values) from
(15)
Step 7: Repeat Steps 4–6 until region 3 is reached.
Step 8: In region 3 the arrival rate to the system is estimated
using (taking into consideration that the number of
users is finite and small and assuming
as [1])
(16)
where
if a success was observed during time slot
otherwise
AL-AMIR AND GURCAN: ARRIVAL RATE ESTIMATION ALGORITHM FOR SINGLE GROUP SLOTTED ALOHA SYSTEMS 389
Fig. 2. Tracking performance ofN by S using the following
parametersN = 120 users,S = 40 users, =
0:2 packets/slot(u ; u ; u ) = ( 1:2;0:4;0:4) for proposed algorithm, with
 = 0:18 (u ; u ; u ) = ( 0:33;0:4;0:4) for Rivest’s algorithm, with
 = 0:5.
Step 9: Calculate a new value of (keeping and
fixed at the initial values) from
(17)
Step 10: Repeat Steps 8 and 9. If number of backlogged
users increases, repeat from Step 3.
The size of the correction determines the speed at which
the estimated arrival rate converges toward the real arrival rate
and can be chosen to be either of fixed or variable size.
IV. SIMULATION RESULTS
Simulation results are obtained by updating and ac-
cording to the changes occurring in the system for a given set
of initial input parameters ( , real number of backlogged
users , total number of users (real) in the system , es-
timated number of backlogged usersand control parameters
, and ). For each time iteration, the number of users that
have new packets to transmit are computed and are added to
the number of users in the backlog state that are waiting to re-
transmit their packets. Users in the backlog state transmit their
packets according to their retransmission probability. At
the end of each iteration, the values of the output parameters
( ) are processed to obtain the different plots pre-
sented.
The control parameters are adjusted using the arrival rate es-
timation algorithms presented by Rivest [2] and the tracking
algorithm proposed in this letter, are shown in Figs. 2 and 3.
Fig. 2 shows the tracking performance of by when the
control parameters are updated using the two algorithms. From
this figure it can be seen that tends to overestimate after
some time when control parameters are updated using the al-
gorithm proposed in [2]. This indicates that the load offered to
the system is not at the optimum value, hence it cannot utilize
the Aloha channel efficiently. For the proposed algorithm,
closely follows once it approaches it, thus causing the of-
fered load to settle at the optimum offered value. Fig. 3 shows
Fig. 3. Tracking performance of by  using following
parameters(u ; u ; u ) = ( 1:2;0:4;0:4) for proposed algorithm, with
 = 0:18 (u ; u ; u ) = ( 0:33;0:4;0:4) for Rivest’s algorithm, with
 = 0:5.
the tracking performance of based on its estimation using
both algorithms. For the algorithm presented in [2] it can be seen
that the large initial value of the arrival rate makes the time taken
for the estimated arrival rate to reach the real arrival rate much
longer. For the proposed algorithm, the precise tracking of the
real arrival rate causes the system to settle at the optimum op-
erating point, thereby achieving maximum throughput. On the
other hand, the algorithm presented in [2] is not suited in esti-
mating the arrival rate over the entire load range.
V. CONCLUSIONS
This letter proposes a new recursive tracking algorithm for
the real arrival rate for Slotted Aloha systems. Each estimate of
the arrival rate is used to update the control parameters of the
system in order to ensure that the operating point of the system
is pushed toward the optimum settling point. Simulation results
verify the effectiveness of the algorithm.
In real time systems, feedback information from the base
station is made available to users in the system at the end of
each time slot, assuming that the channel propagation delay is
negligible. This is possible by keeping the packet slot duration
smaller than the time slot to overcome these propagation delay
p oblems occurring due to users being at different distance away
from the base station.
REFERENCES
[1] S. C. A. Thomopoulos, “A simple and versatile decentralised control for
slotted Aloha, reservation Aloha and local area networks,”IEEE Trans.
Commun., vol. 36, pp. 662–674, June 1988.
[2] R. L. Rivest, “Network control by Bayesian broadcast,”IEEE Trans.
Inform. Theory, vol. IT-33, pp. 323–328, May 1987.
[3] M. K. Gurcan, A. Al-Amir, and M. Hijazi, “A sequential method for con-
trol parameters in stabilised ALOHA systems,”IEEE Commun. Lett.,
vol. 3, no. 5, pp. 148–150, May 1999.
[4] L. P. Clare, “Control procedures for slotted ALOHA systems that
achieve stability,” inProc. ACM SIGCOMM ’86 Symp. Commun., Arch.
& Protocols, Stowe, VT, Aug. 1986, pp. 302–309.
[5] Y.-W. Leung, “Mean power consumption of artificial power capture in
wireless networks,”IEEE Trans. Commun., vol. 45, pp. 957–964, Aug.
1997.


Arrival Rate Estimation Algorithm for Single Group Slotted ALOHA Systems

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Arrival Rate Estimation Algorithm for Single Group Slotted ALOHA Systems

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