Distributed Timeslot Allocation in mMTC Networkby Magnitude-Sensitive Bayesian Attractor Model
Tatsuya Otoshi

Masayuki Murata, Hideyuki Shimonishi, Tetsuya Shimokawa

Graduate School of Economics
Osaka University
Osaka, Japant-otoshi@econ.osaka-u.ac.jp

Graduate School of Information Science and Technology
Osaka University
Osaka, Japan{murata, t-shimokawa}@ist.osaka-u.ac.jp, shimonishi.cmc@osaka-u.ac.jp

Abstract—In 5G, ﬂexible resource management, mainly by basestations, will enable support for a variety of use cases. However,in a situation where a large number of devices exist, such as
in mMTC, devices need to allocate resources appropriately inan autonomous decentralized manner. In this paper, autonomous
decentralized timeslot allocation is achieved by using a decisionmodel for each device. As a decision model, we propose an extensionof the Bayesian Attractor Model (BAM) using Bayesian estimation.
The proposed model incorporates a feature of human decisionmaking called magnitude sensitivity, where the time to decisionvaries with the sum of the values of all alternatives. This allows thenatural introduction of the behavior of making a decision quickly
when a time slot is available and waiting otherwise. Simulationbased evaluations show that the proposed method can avoid timeslot conﬂicts during congestion more effectively than conventional
Q-learning based time slot selection.
Index Terms—Bayesian Attractor Model, Timeslot Allocation,
mMTC, 5G NR, Value-based Decision Making

I. I NTRODUCTION
In recent years, 5G coverage areas have been gradually
expanding, and 5G is on the verge of becoming widespread.
The main feature of 5G is that different communication qualities
can be provided simultaneously by deﬁning the communication
quality to be provided for each communication requirement of
the service [1]. Particularly representative are eMBB (enhanced
Mobile Broadband), URLLC (Ultra-Reliable and Low LatencyCommunications), and mMTC (massive Machine Type Communications). To realize these different communication types
simultaneously on the same communication infrastructure, different technological elements have to be assembled, and this is
likely to be the case for future communication schemes beyond
5G.
One of the key features in satisfying these different communication requirements simultaneously on the same substrate istime slot allocation. 5G divides radio resources into time andfrequency and allocates them accordingly to satisfy different
communication requirements simultaneously. Basically, in thecase of 5G, the base station centrally determines the time slot
and frequency, and the terminal often follows accordingly [2],[3]. However, in some cases, such as communication start timeand mMTC, where a large number of terminals are expected,
random access is used, where terminals select their time slots
autonomously.

In particular, in mMTC, there are many other devices, and
it is necessary to select a timeslot that does not collide withother devices based on limited information about the movementsof all devices, which is not known. A simple way to avoid
collisions would be for all devices to transmit sparingly [4], but
excessive sparing of transmission will reduce overall throughput.
A method [5] that uses reinforcement learning to determine
the time slots to transmit has also been proposed, but it does
not include control over waiting to transmit and is limited tosituations where each device only sends a certain number ofpackets.
In this paper, we propose an autonomous decentralized timeslot allocation scheme, including transmission latency, inspiredby the human decision-making property [6]. Magnitude sensitivity refers to the human tendency to make choices when
there are multiple alternatives with different values and isinﬂuenced by the sum of the values of the alternatives (called
magnitude). It allows waiting to make a choice for a better
choice in the future [7]. Such properties are also known toplay an important role in consensus building in group decisionmaking [8]. Autonomous decentralized timeslot allocation canbe viewed as consensus building regarding the timeslot to be
used by each device, which is a problem compatible withmagnitude sensitivity.
Our research group has applied a decision model called theBayesian Attractor Model (BAM) to several network control
problems (e.g., [9]). BAM is a model that takes observed
information as input, updates the internal state of the BAM
itself, and outputs the results of the decision process. The
original BAM does not take values as input, but rather instances
for typical patterns, but in this paper, we extend this to avalue-based decision model. As we will show later, there issome magnitude sensitivity in the BAM itself when extended tovalue-based decision-making. In addition, its similarity to otherdecision models makes it easy to incorporate models with othermagnitude sensitivities.
Due to the applicability of such BAMs to network control
and their compatibility with magnitude sensitivity, this paper
proposes a method for autonomous decentralized timeslot allocation using an extended model of value-based BAMs. In thismethod, each terminal independently determines the timeslot to
be used. At this time, the terminal also considers the option ofnot making a decision, i.e., waiting for transmission, according
to the magnitude sensitivity. After selecting a timeslot, the basestation provides feedback to the terminal on the availability ofthe timeslot, and the terminal updates the value of the choice
and moves on to the next choice. Through the evaluation bysimulation, we show that the proposed method can determine
when to wait for a decision and when to make a decision,
depending on the situation, and make an appropriate choice.
The remainder of this paper is organized as follows. In
Section II, we propose a new model of magnitude-sensitive
decision-making that extends the Bayesian attractor model. In
Section III, we present an application of the proposed modelto achieve distributed timeslot allocation in a random accessenvironment at mMTC. In section IV, we evaluate the behaviorof the proposed model. Finally, in section V, we summarize anddiscuss future work.
II. M AGNITUDE -S ENSITIVE BAYESIAN ATTRACTOR M ODELIn this chapter, we propose a new model of decision-making
that is magnitude sensitive and highly applicable to engineering.
First, we introduce the basic model, BAM. Our research group
has applied BAM to various network control problems, andBAM is a model with high applicability. However, since theoriginal BAM is a feature-based classiﬁcation model, which
is different from the value-based decision-making targeted inthis paper, we introduce an extended BAM for value-based
decision-making. We then propose a new model that integrates
the conventional magnitude sensitivity models, LCA and IDN,
into the value-based BAM.
A. Value-based BAM
1) Bayesian Attractor Model: BAM [10] is a model of brain
decision-making that involves the process of updating internal
states based on observations, and making decisions based on the
updated internal states.
The original BAM models the process of reafﬁrmation, in
which features are used as input and representative values are
updated by comparing them to representative values bound toa previously given choice, called an attractor. The speciﬁc state
update is performed by Bayesian updating based on observed
values, using the relationship between attractors and representative values, and the endogenous dynamics of the state as a
generative model. The generative model is as follows
zt=

f (zt−1 ) + qwt

(1)

xt

=

M σ(zt ) + svt(2)

where xt is the observed value, zt is the internal state, andwt , vt is the noise. Eq. (1) is an expression for the internalvariation of the state, where f is the Hopﬁeld dynamics with K
attractors ϕ1 , · · · , ϕK . Eq. (2) is an expression for the relation
between representative values and attractors, where M is amatrix of representative values µi corresponding to the attractor
ϕi and M = (µ1 , · · · , µK ). The σ(zt ) is an element-wise
sigmoid function. Also, q, s are the parameters for the magnitudeof each noise term, called dynamic uncertainty and sensory
uncertainty.

2) Values as Observations and Representatives: As noted
above, the original BAM is a recognition model and does notaddress the value of alternatives. In this paper, the observed andrepresentative values are changed to value-based values of thealternatives in order to make value-based decisions in the BAM.
Let Vi,t be the value of choice ϕi at time t, and let BAM
obtain a value estimated value v̄i,t through reward feedback
information. This sequence of value estimates is the informationthat BAM can observe at time t, and the observed value as value
is deﬁned as xt = (v̄1,t , · · · , v̄K,t ).
In value-based decision-making, we need to ﬁnd the highestvalue alternative. To handle this in the recognition schemeof BAM, the representative value µi of the choice ϕi is theobserved value such that the choice is of maximum value.
That is, using the standard value v̄ and the maximum valuev̄max = max{v̄1,t , · · · , v̄K,t }, we determine the representativevalue as follows(i)µi = (v̄0 , · · · , v̄0 , v̄max , v̄0 , · · · , v̄0 ).

(3)

In this paper, v̄0 = 0. The representative value closest to theobserved value xt is µmax , corresponding to the option withthe largest value ϕmax . Therefore, it is possible to choose theoption with the largest value as the scheme to recognize theobserved value.
In a normal BAM, the observed and representative values
are often normalized, but here the values are used as they arewithout normalization. This leads to a large deviation from theleft-hand side of Eq. (2) in the case of large values. As a resultof Bayesian estimation, which attempts to eliminate this bias,
the magnitude sensitivity property is at work, shifting the updateof zt to a larger value.
3) Decision Process: In the original BAM, the alternative isselected for which the conﬁdence P (z t = ϕi |x1..t ) is above
the threshold. On the other hand, in a value-based BAM, theconﬁdence level is determined by the relative proximity to thelargest value, so it is not appropriate to set a threshold that is
an absolute standard for the conﬁdence level.
Therefore, ϕf irst and ϕsecond are selected if the ratio of the
conﬁdence level of the top two options ϕf irst and ϕsecond isgreater than some threshold θ. In other words, ϕf irst is selected
if the following conditions are met, otherwise we wait for thedecision.
P (z t = ϕf irst |x1..t )>θ
(4)
P (z t = ϕsecond |x1..t )where x1..t is a sequence of observations from time 1 to timet.
B. BAM-LCA
LCA is a model of magnitude sensitivity decision-making
based on the diffusion model [11], [12].
In the diffusion model, the internal state is updated by adrift term and a noise term, and decisions are made when theinternal state exceeds a threshold; in LCA, magnitude sensitivity
is expressed by varying the drift term in the diffusion modelin a stimulus size-dependent manner. Speciﬁcally, the internal
state Xi for each option is updated according to the following
equation.
Xi,t+1 = Ii (t) + (1 − γ)Xi,t − β

∑

Xj

(5)

j̸=i

where Ii (t) represents the stimulus magnitude, and γ, β are
the parameters. The ﬁrst term indicates that the more valuable
the choice, the larger the movement of the state. The second term
indicates self-activation, and the third term indicates suppressionof other choices by the active choice. In the steady state, only
one higher-value option is active, similar to BAM.
In value-based BAM, the deﬁnition of M, xt in eq. (2) indirectly causes an internal state update according to the magnitudeof the value. However, it is s that controls the effect of Eq. (2) on
the state update. The smaller s is, the more strictly x follows
the equation, and the more deterministically x updates in theBayesian update.
Therefore, by varying s as a function of magnitude, the valuebased z update can be more directly controlled. In this paper, s
is varied by the reciprocal of the magnitude as follows.
s=

s0
Vt

(6)

∑
where s0 is the reference sensory uncertainty and Vt = i v̄iis the magnitude. Assume s0 = 1 unless otherwise noted.
C. BAM-IDN
IDN is also a model of decision-making with magnitudesensitivity based on the diffusion model. Speciﬁcally, the noise
term ξi,t is given by
ξi,t ∼ N (0, πv̄i2 + σ 2 )(7)where π, σ are parameters. When the value is large, the varianceof the noise term is larger and the diffusive state change isaccelerated. As a result, the time it takes for the state to cross thethreshold is reduced and decisions are made faster in situations
where the value is large.
In the BAM, the parameter that controls the magnitude of thenoise term is q. Therefore, by varying the magnitude of q in avalue-dependent manner, an effect similar to that of IDNs canbe realized. In this paper, the generating distribution of q andthe gamma distribution are given by the following equation, sothat the expected value of q depends on the magnitude.

q∼E[q] =

Γ(kΓ , θΓ )(8)kΓ θΓ = πVt2 + σ 2(9)where kΓ , θΓ are the parameters of the gamma distribution
and satisfy Eq. (9). However, since there remains 1 degree ofπV 2 +σ 2
freedom in the parameters, we shift θΓ and set kΓ = tθΓ .
In this paper, θΓ = 6, π = 1, σ = 0 unless otherwise noted.

III. D ISTRIBUTED T IMESLOT A LLOCATION WITH
VALUE - BASED BAM
Autonomous decentralized timeslot allocation is performed
using the value-based extended BAM model proposed in theprevious chapter. In this method, each device independently
makes decisions based on the value-based BAM model andselects a timeslot to transmit. Appropriate timeslots are selectedby receiving value-based feedback from the base station about
timeslot congestion. First, the assumed system model is described. Then, the speciﬁc application of value-based BAM totimeslot decision-making is described.
A. System ModelIn this paper, we assume a system model that is nearly
equivalent to that in Ref. [5]. The main change is that in thereference [5], the number of packets to be sent by the device isassumed to be ﬁxed at the beginning, whereas in this paper, thepackets to be sent by the device are assumed to be generated ona continuous basis. This is considered to be more compatible
with mMTC use cases, such as sensor networks.
Assuming an mMTC network, all N devices send data to thebase station. A subframe consists of K timeslots.
Only if only one device chooses to send a packet for a given
time slot, the packet transmission is considered successful. If
two or more devices are in use in the same time slot, there
is a collision and the packet transmission will fail. If thetransmission fails, it is up to the application to decide whether
to retransmit the packet, which is described by the packet
generation process. To simplify the analysis, physical channel
losses such as multipath fading are assumed to be negligible.
Before the start of the next subframe, the base station feeds
back to the devices the congestion in the time slot quantized inb bits. When the number of devices is less than the subframe
length, it is more efﬁcient to provide feedback on a perdevice basis, but when the number of devices exceeds thesubframe length, it is more efﬁcient to broadcast information ona per-timeslot basis. mMTC mainly assumes a large number ofdevices, so it is more efﬁcient to broadcast information on a pertimeslot basis. Feedback is assumed to broadcast the congestion
level of each timeslot.
B. Timeslot Allocation
Each device independently makes decisions about timeslot
selection and transmission waiting through value-based BAM.
Each device updates its internal state based on feedback fromthe base station and incorporates it into the next decision.
The following sections describe the attractors and how to map
decisions to timeslot selection in value-based BAM.
1) Attractor: K attractors ϕ1 , · · · , ϕK corresponding to the
K timeslots in the subframe are prepared. The attractor ϕK
corresponds to the selection of the Kth time slot.
2) Value of Timeslot: The value-based BAM estimates thevalue of each attractor from the feedback and uses the estimatedv̄k,t as the observed value to update the internal state.
In the feedback information, a value of 1 is given for a
successful transmission, and a negative value is given for acollision, discretized by b bits of congestion in the timeslot.
More speciﬁcally, the value of timeslot k at time t is given by{
vk,t =+1 if transmission succeedsN−Mb ( Nk,t ) otherwise(10)

where Mb (x) is the discretized value of congestion in b bits,
and B = 2, Mb (x) ∈ 0.25, 0.5, 0.75, 1. Let Nk,t denote thenumber of devices that have selected timeslot k at time t. We
assume broadcast feedback in this paper.
The feedback value reﬂects instantaneous congestion in a
subframe. Since this value includes temporary congestion, we
try to get a more stable understanding of congestion by using
time-smoothed estimates. That is, as feedback is received, theestimates are updated as follows
v̄k,t = (1 − α)v̄k,t + αvk,t(11)where α is the smoothing parameter. Unless otherwise noted,
α = 0.3 is used in this paper. This corresponds to the Q value.
In the case of Q-learning, updating all time slots is negative
because Q-value synchronization causes congestion in a timeslot. In value-based BAM, on the other hand, the BAM itself
has the consistency of decision-making as an internal state, soeven if value updates are synchronized, each device can remain
in a different selection state. Therefore, for value-based BAMs,
the update of v̄k,t is performed for all timeslots.
3) Decision: In value-based BAM, the maximum conﬁdence
selection is made if the ratio of the conﬁdence levels for the
top two choices exceeds the threshold, otherwise, a decision
is awaited. If the attractor ϕk is selected, it selects timeslot k
and attempts to transmit the packet. On the other hand, if nodecision is made in this subframe, no timeslot is selected andpacket transmission waits until the next subframe.
IV. E VALUATION
The proposed model and its application to autonomous decentralized timeslot allocation are evaluated by simulation. In
this chapter, we conﬁrm that the proposed model is magnitudesensitive and investigate how the selection tendency is affectedby magnitude. Then, we evaluate by simulation the behavior of
the proposed model when applied to timeslot allocation under
the assumption of mMTC.
A. Magnitude Sensitivity of the Model
Using a numerical example, we conﬁrm the magnitude sensitivity of the value-based BAM described in section II. By
directly changing v̄1,t , · · · , v̄K,t , the inputs in the value-based
BAM, we can see how the behavior of the model changes withthe magnitude of the value.
Figure. 1 shows the decision time and accuracy for different
magnitudes. In the ﬁgure, ”BAM” represents value-based BAM,
”IDN” represents BAM-IDN, and ”LCA” represents the resultsof BAM-LCA. accuracy indicates the percentage of time thedecision at each time point is consistent with the correct answer.
The ﬁgure shows that all models show magnitude sensitivity,
which means that the speed of decision-making increases as themagnitude increases. This tendency is particularly pronounced
for BAM-IDN and BAM-LCA, which are magnitude-sensitive
extensions of value-based BAM.
Accuracy is also found to improve with increasing magnitude.
In general, it seems that the earlier the decision is made, thelower the accuracy. However, in the present case, the total
number of correct answers increases with the earlier decision,
because the state of waiting for a decision is automaticallyassumed to be incorrect.
BAM-LCA has better decision time and accuracy than othermodels, but this may be due to the magnitude improvement
in decision behavior. Originally, in BAM, the decisive action
brings us closer to the prepared choice, so by emphasizing thisbehavior, we quickly arrive at the correct answer.
B. Simulation of Timeslot Allocation in mMTC
To conﬁrm the effectiveness of autonomous decentralized
timeslot allocation in mMTC with magnitude-sensitive BAM,
we evaluate the proposed method by simulation. In the simulation, each device continuously sends packets to the base station,and the proposed method is used to select the timeslot in which
the packets are sent.
1) Setting: As described in section III-A, N devices occupy K timeslots simultaneously. A device decides whether totransmit a packet per subframe and, if so, chooses a timeslot.
The number of timeslots per subframe is determined by thenumerology µ, where K = 2µ . Unless otherwise noted, µ = 4.
Since we wanted to check the response to changes in spatialload, N was allowed to vary during the simulation.
Before the next subframe starts, the base station gives feedback to the device. The feedback is a discretized value ofcongestion in b bits. We chose b = 2 to match the settings
in Ref. [5].
Collision and efﬁciency are used as performance metrics.
collision is the ratio of the number of timeslots in whichcollisions occur out of K timeslots. Efﬁciency is the ratio of thenumber of timeslots in which communication succeeds out of
K timeslots. Therefore, the smaller the collision and the higher
the efﬁciency, the more appropriate timeslot allocation can beachieved by avoiding collisions.
2) Behavior of Each Model: To verify the behavior of eachmodel, we ﬁrst check the time slot allocation for value-based
BAM and BAM-IDN and BAM-LCA, respectively. The numberof devices is assumed to be N = 10 to see if collision avoidance
can be achieved in a situation where collisions are in principle
avoidable.
Figure 2 shows the time series of collisions resulting fromthe time slot allocation when using each model. One step on thehorizontal axis represents one subframe and the vertical axis is
the percentage of collisions that occurred during that subframe.
”BAM”, ”IDN” and ”LCA” represent the results of value-based
BAM, BAM-IDN, and BAM-LCA, respectively.
The ﬁgure shows that BAM-LCA achieves collision avoidance. Value-based BAM also shows a gradual decrease incollisions, but not to the point of collision avoidance. In addition,
BAM-IDN shows an increase and decrease in collisions due tonoise-induced changes in decision-making.

1.0

100

0.8

method

BAM
IDN
LCA

50

accuracy

decision time75

method

BAM
IDN
LCA

0.6
25

0

0.4
10

20

magnitude

30

40

10

(a) Decision Time20

magnitude 3040(b) Accuracy

Fig. 1. Magnitude Sensitivity of Each Model’s Decision Time and Accuracy

ACKNOWLEDGMENT

0.16

This work was supported by MIC under a grant entitled ”R&D
of ICT Priority Technology (JPMI00316)”.

collision

0.12

R EFERENCES

0.08

type
BAM

0.04

IDN
LCA
0

25

50

75

100

time
Fig. 2. Timeseries of Collision for Each Value-based Decision Model

V. C ONCLUSION
In this paper, magnitude-sensitive decision models are applied
to achieve autonomous decentralized timeslot allocation for
mMTC networks. Based on these models, we proposed themagnitude-sensitive BAM, BAM-LCA, and BAM-IDN, which
are extensions of BAM. Using these models, we proposed amethod in which each device autonomously and decentrally
selects an appropriate timeslot through feedback that reﬂects thevalue of timeslot congestion. Through simulation-based evaluation, we showed that BAM-LCA can avoid timeslot collisions
compared to other methods. We also showed that BAM-LCA ismore efﬁcient in timeslot utilization, especially when the numberof devices is large.
Future work includes a comprehensive performance evaluation in various settings and a study of how to switch to more
appropriate decisions based on the number of devices.

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