INTRODUCTION
Traffic load in wireless communication networks is subject to time and regional variations, which leads to variations in the utilization of spectrum resources. Therefore, radio spectrum can be underused by one system and may experience a shortage with another (Leaves et al., 2004; Sharif et al., 2005; Sharif and Fan, 2006; Xia et al., 2007). As indicated by the Spectrum Policy Task Force (FCC, 2002) and by other reports such as the measurements conducted from January, 2004August, 2005 for frequency bands below 3 GHz (Cordeiro et al., 2005; McHenry, 2005), the average spectrum usage is only about 5.2%. Interestingly, it also reveals that heavy spectrum utilization often takes place in unlicensed bands while licensed bands often experience low or medium utilization (e.g., TV or cellular bands, respectively). Therefore, for the dynamic spectrum allocation the spectrum management policies need revision.
Most of the public debates on the spectrum management policies usually break
out into two dissimilar approaches, Commons and QuasiPrivate Property (Cooper,
2006).
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The Commons spectrum approach is one where all systems
have equal rights to access the spectrum, e.g., today’ unlicensed
bands. Unlicensed spectrum bands have spurred tremendous innovation, productivity
and led to greater spectral efficiency as compared to licensed bands.
However, there are some challenges related to implementation of the Commons.

• 
The spectrum as QuasiPrivate Property encourages spectrum sharing between
licensed donor and rental systems where the rental systems must not cause
any harmful interference to the donor system. However, in a quasiprivate
property approach, the change in legacy systems is needed that is, the donor
systems need additional components which will act as a gatekeeper. 
Due to the high spectrum scarcity in unlicensed bands and due to the fact that the significant revolutions in radio spectrum usage have originated in the unlicensed bands, the spectrum as a Commons is the focused spectrum sharing environment.
To realize the potential benefits of spectrum sharing, there are some challenges to overcome. One out of those is the possible tendency of greed in individual systems. In unreliable radio environments such as the unlicensed frequency bands or Commons, contentionbased protocols are used for wireless communication due to their irregular and unpredictable interferences. Therefore, to support timebounded traffic with a certain Quality of Service (QoS) is extremely difficult (Mangold et al., 2004). It is because in contention based protocols there is no surety regarding availability of free spectrum. In addition to that, there may be a tendency of greed in individual systems to hold the spectrum band more than their requirement to avoid delay of accessing the band again, in particular under heavy traffic load. This may cause a great deal of problems for real time applications of all other systems waiting for transmission. As a result, these individual systems may waste spectrum to improve their own performance at the expense of overall spectral efficiency and performance degradation for other systems. If this kind of behavior happens frequently, which is quite possible under heavy traffic load, then the benefits of dynamic spectrum allocation among multiple radio systems may disappear, or may turn out to be more critical for real time traffic. With the plethora of new real time applications emergence, the tendency of greed becomes more critical. Therefore, the awareness of the AWT and the corresponding controlling factors, is crucial for a system to manage the time sensitive traffic.
To minimize the AWT and the impact of greed Greed Elimination Model is presented and analyzed. In this study, detailed analysis and simulation are performed to investigate the controlling factors for AWT. Furthermore, some possible improvements to minimize the said problem are also presented.
GREED ELIMINATION MODEL AND ASSUMPTIONS
Our model is based upon the opportunistic spectrum sharing, as it is one of the most suitable way to share the spectrum, when the spectrum is taken as a Commons. Suppose under opportunistic spectrum sharing scenario, k represents the average number of systems competing for free subbands at any instant and while waiting for transmission, all the blocked systems are queuing up their data independently. Another basic assumption made is the probability of collision while accessing the subband is considered negligible, as it may happen because the subband assessment is not instantaneous.
For minimizing the AWT and the impact of greed, our model put an upper limit
on the transmission duration in conjunction with the Distributed Coordination
Function (DCF). That is, any system can transmit its queued data uninterruptedly,
no longer than the Q duration of time. As the DCF is based on Carrier Sense
Multiple Access with Collision Avoidance (CSMA/CA) and binary exponential backoff,
therefore, brief introduction for the same can be viewed as follows. Before
each transmission, a system have to monitor the subband for M units of time.
If during this interval, the power on the subband never exceeds a predefined
deference threshold, then the system has the permission to transmit. If the
power exceeds the deference threshold the system defers a random amount of time,
called the deference interval D_{L} and repeats the monitoring procedure
after that.
where,
The range [M, L] from which the uniformly distributed deference interval is chosen is doubled each time a system detects subband busy. The lower limit is fixed at some suitable value M, while the upper limit is initially l_{min} and step by step reaches at l_{max} after some unsuccessful tries. After approaching l_{max} the range for deference interval is kept constant indefinitely.
PERFORMANCE ANALYSIS
Consider at any time t, the jth system wins the contention after some unsuccessful
tries. At time t+Q system j stops transmission and draws a deference delay from
a uniform distribution D_{L }~ U(M, l_{min}). The time
intervals from the instant the subband becomes clear (i.e., t+Q) until the blocked
systems are scheduled to monitor the subband are random variables, denoted by
R_{i,} where, R_{i }≠ R_{j}, as shown in Fig.
1.
According to the Fig. 1, who will win the contention for
transmission in the next cycle, depends upon the remaining deference time R_{i},
the lesser the better. As any random variable R_{i} represents the residual
life for sensing the carrier again, therefore, the Residual Life Theorem (Kleinrock,
1976) can be applied for the distribution of R_{i}, as given below,
where:
is the cumulative distribution function of deference interval D_{L}
and E[D_{L}] is the mean of D_{L}.
Deference process after approaching at maximum limit l_{max}
becomes a renewal process. Under heavy traffic load, it is quite reasonable
to assume that the renewal process starts during the first transmission period
and that it will be in steady state at the end of the first busy period. Therefore,
Eq. 2 can be used for all blocked systems.

Fig. 1: 
Residual life and competition for next transmission cycle 
The distribution function of deference delay for the currently transmitting
system j can be written as,
The probability of any blocked system to win the contention for transmission
in the next cycle can be obtained by using joint probability function for two
variables. Figure 1 shows that any system i with minimum R_{i}
out of all blocked systems will compete with the currently transmitting system
j for transmission in the next cycle.
where For equally probable, independent and identically distributed random variables,
we have:
where, F(r) is the cumulative distribution function of R_{1}. Applying
this to Eq. 4, we can get:
It is quite clear that any of the (k1) blocked systems can win the contention
with equal probability. Thus the probability that a particular blocked system
will transmit in the next transmission cycle can be written as:
Therefore, the average number of transmission cycles an arbitrary system will
wait can be obtained from,
In order to find the average waiting time, there is a need to get average cycle
period, because all cycles are not the same in length. Each cycle consists of
two parts, the transmission time Q and the idle time I, which can be seen in
Fig. 2, where Q is a constant, but I depends upon residual life of random
variables R_{i}. Therefore, the average idle time E[I] is needed first
which can be calculated as given in Eq. 9. The average waiting
time for any blocked system can be written as,
where,

Fig. 2: 
Length of transmission cycles 
RESULTS AND DISCUSSION
In the simulation and analytical studies performed, it is assumed that the
free band monitoring time is 50 μ sec and the upper limit for the deference
interval can reach the maximum 12 m sec. Furthermore, in each transmission cycle
there are 10 m sec for burst transmission, whereas, the burst may consist of
a large number of short packets, provided that the interpacket gap is less
than 20 μsec. Keeping in mind the time sensitive applications, the significance
of the model considered can be seen in results obtained, i.e., the average waiting
time is raising very slowly with the increase in the number of systems. The
growing number of systems is actually accommodated with the reduction in the
average transmission time which is also apparent from the simulation results.
The initial upper limit for deference interval also plays an important role
for the average waiting time, a slight increase in the initial limit reduces
the average waiting time considerably. The simulation is performed for three
different values of initial upper limit for deference interval, 0.5, 0.75 and
1, as shown in Fig. 3.
Figure 4 presents the simulation and theoretical results
obtained that the analytical results fit well with the simulation results. The
difference between the simulation and analytical results is approximately 4%
or less. It is apparent from the simulation results that lowering the upper
limit of transmission duration reduces the average waiting time sharply (Fig.
5).
It is shown here that the AWT is a function of upper limit on the transmission
duration Q and the initial upper limit for deference interval l_{min}.
However, the ATT depends a lot upon the average number of systems K, competing
for spectrum bands at any instant of time. As far as the AWT is concerned, it
increases very slowly with the increase in the number of systems competing for
spectrum band, which is the beauty of the model considered. Furthermore, it
is quite clear from the results obtained that reducing the maximum transmission
duration decreases the average waiting time for all blocked systems.

Fig. 3: 
Simulation results for AWT and ATT as a function of the
number of systems competing for spectrum 

Fig. 4: 
Simulation and numerical results for AWT as a function of
the number of systems competing for spectrum 
For example,
for Q = 10, 5, 1 m sec, the AWT goes to 220, 113, 15 m sec, respectively, for
a particular set of parameters. But, reducing the transmission time causes a
noticeable wastage of spectrum resources.

Fig. 5: 
AWT as a function of upper limit on transmission duration 
It is because reducing the transmission
time increases the number of occurrences for nonnegligible probability of the
idle time and consequently there is an increase in the overall spectrum wastage. Similarly, the initial upper range l_{min} for deference interval
also plays an important role for the reduction in the average waiting time,
However, there is a possibility of slight spectrum wastage in terms of a comparatively
longer idle time. Therefore, by knowing the impact of controlling factors as
discussed earlier, an optimal solution for the upper limit on the transmission
duration and for the initial upper range of deference interval can easily be
found through the tradeoff between the acceptable average waiting time for the
time sensitive applications and the overall spectrum utilization.
CONCLUSIONS
To minimize the average waiting time and the impact of greed in the scenario
where the spectrum is taken as a commons, greed elimination model is presented
and analyzed. In this study, keeping in mind the importance of the AWT for time
sensitive applications, the analytical and simulation studies are performed
to investigate the controlling factors for AWT, which is helpful for those systems
who do not want to invest a lot for change in legacy systems. The results obtained
present the impact of transmission duration, the average number of systems competing
for spectrum and the initial upper limit for deference interval, upon the AWT
and the ATT. With some trade off in these parameters, the acceptable AWT for
the time sensitive applications can easily be found.
ACKNOWLEDGMENT
This study was supported by the National Science Foundation of China (NSFC) under the grant No. 90604035 and No. 60440420451 (Two Base Project).