1.1 BitTorrent

BitTorrent is a P2P application used to facilitate the download of popular files. The main idea is to divide the files into several pieces called chunks. To download a file, peers exchange these chunks following some rules.

The BitTorrent protocol differentiates two types of peers: leeches, which are peers that have a part of the file or not data at all, and seeds, which are those peers having downloaded the complete file and remaining in the system to share their resources. Both leeches and seeds cooperate to upload the file to other leeches. Whenever a peer joins the system with the objective of downloading the file, it contacts a particular node called tracker which has the compete list of peers that have part or the complete file. Then, the tracker returns a random list of potential peers that might share the file with the arriving peer. At this point, the downloading peer contacts the peers on the list and establishes which chunks it is willing to download from each peer it is connected with. The decision of which peers to upload to depends basically on how much chunks have downloaded the peers in question. Hence, the peers that have downloaded the most are the ones that have priority over the peers that do not share their chunks, discouraging the free riders (peers that only download but do not share their data).

1.2 Contributions

The rest of the paper is organized as follows. Section II reviews the previous works related to mathematical modeling and incentives in P2P networks. Then, section III specifies the system model and assumptions, including the analysis of the system’ conditions where the priority schemes are suited, the models used in this work and the simulation model. Following this, section IV presents the relevant numerical results in order to evaluate the impact on the distribution of the transfer rates and the priority based mechanisms proposed in section III. The article concludes with a summary of our conclusions and contributions.

3.1 Single Population Models

The mathematical models presented in this section are a simplification of a BitTorrent-like network. Indeed, the real system is much more complex due to the numerous parameters and details involved in the file searching, node selection, and chunk selection procedures, among others. Specifically, the mathematical models assume that whenever a peer arrives, it is immediately served by the rest of the peers in the network, i.e., the total uploading bandwidth is uniformly distributed among all users in the system. Another important assumption is that all the peers in the system cooperate, or that all users have chunks that are of interest for the rest of the peers. This is a very important problematic that has been extensively studied before (for instance in [¹³], the effectiveness of the file sharing is studied with a parameter η that represents the probability that a peer is interested in the chunks of other peers, similar also to the work presented in [²]) and therefore we do not further study this issue. Finally, one more detail that is not represented in the mathematical models is the fact that the BitTorrent system only allows for a maximum of 4 active uploading connections for any peer. An additional peer is selected randomly in order to better distribute the file in the network under the optimistic unchoking policy. Next we describe the rest of the assumptions and system parameters.

New peers arrive to the system according to a Poisson process having rate λ and are labeled as leeches. All peers have the same uploading rate μ and the same downloading rate c, c>μ. A single file download is considered. At any given time there is at least one seed in the system. All peers have complete knowledge of the system, i.e., all peers know which chunks has any other peer and peers always cooperate to upload data if they have available bandwidth. As such, if the number of leeches and seeds is sufficiently high, all leeches download the file at the maximum download rate c. However, when there are not enough peers in the system, the leeches download at rate μ(x+y), where x is the instantaneous number of leeches and y is the instantaneous number of seeds. With respect to [²], we simplify the model by assuming that the efficiency η defined in that paper is 1. From the previous description, the evolution with time x and y satisfies:

Observe that in these equations, stated in [²], the condition y(t)≥1 is not considered. One way to do it is to denote by y(t) the number of seeds in the system other than the fixed ones, at time t, and to replace min⁡(cx(t),μ[x(t)+y(t)]) by min⁡(cx(t),μ[x(t)+y(t)+1]), but we kept the equations as in the original paper, for simplicity. Remember that these differential equations make sense for high population values.

In [²], this dynamical system is analyzed and its equilibrium point, denoted by (x¯,y¯), is studied. The equilibrium is obtained by solving the system {x′(t)=0,  y′(t)=0} and by replacing x(t) by x¯ and y(t) by y¯. The result is:

where β=(max⁡{1/c, 1/μ−1/γ})−1..

Consider now the evolution of the pair W(t)=(L(t),S(t))≥(0,1), where L(t) and S(t) are respectively the number of leeches and seeds at time t. With all the previous “exponential assumptions”, W(t) is a continuous time homogeneous Markov chain, with initial state W(0)=(0.1). Starting from state (n,m), n≥0 and m≥1, the transition rates are as follows: λ to state (n+1,m), nθ to (n−1,m) (for n≥1), min⁡(cn,μ(n+m)) to (n−1,m+1) (for n≥1), and (m−1)γ to state (n,m−1) (for m≥2).

This Markov chain is analyzed using simulation. Concerning its stability, it is easy to see that it is always stable by seeing the model as a tandem of infinite server queues, the first one containing the leeches, the second one the seeds. The second one is a simple ./M/∞ queue. The first one is a more complex ./M/∞ queue, with a service rate depending on the state of the whole system. Stability follows by lower bounding the service rate at the first queue by μL(t), for instance, and by observing that the resulting model is a Jackson network composed of two ./M/∞ queues, which is always stable.

For the priority scheme, it is important to identify the system’s condition where the bandwidth is scarce due to the low number of peers. Then, numerical values of the number of seeds and leeches in equilibrium are obtained from both the Markov model and the fluid model. It can be seen from Fig. 1, that the Markov model (referred as simulation) achieves a very good match to the closed expressions presented in (3), when cx<μ(x+y)). This condition of abundance (also referred in this document as maximum capacity), is achieved when there is a high number of leeches and seeds in the system, i.e, for θ and γ small and λ high. The values of c and μ are not varied since these parameters are usually defined in advance and in general do not depend on the user behavior.

Fig. 1 Average Number of Leeches and Seeds for Different Arrival and Departure Rates 

In Fig. 1, we can see that when the departure rate of leeches is high, both the number of seeds and the number of leeches is low. This is because most of the peers that enter the system leaves in a small amount of time and there are almost no leeches that convert into seeds. Conversely, when the departure rate of seeds is high, the number of leeches does not decrease since there are still other leeches in the system who share the file with the rest of the peers. The number of seeds does not decrease as drastically as in case of high values of θ, since there are still leeches becoming seeds. More importantly, when γ≥0.004, the system has a high probability to be in penury conditions. Hence, for the introduction of the priority scheme, the value γ=0.001 is chosen to represent abundance conditions whereas the value γ=0.005 is chosen to represent penury conditions.

3.2 Two Population Model

For the two population model, the behavior of two types of users is considered: cooperative leeches (also called high tolerance leeches) and defeat leeches (also called low tolerance leeches). For the former, they are peers that arrive to the system and usually stay throughout the entire download procedure. In other words, they have a high tolerance for download latency. Therefore, their departure rate (θc) is lower than the download rate at the maximum capacity, i.e., c>θc. For the latter, they are peers that arrive to the system and have very little tolerance to download latency. Hence, even if their departure rate is lower than the download rate at the maximum capacity, it is just lower. In view of this, we consider the next set of values: θc=θ and θd=0.9c. Cooperative leeches arrive with rate λc and defeat leeches arrive with rate λd.

According to this, the fluid model presented above is extended for the two types of leeches as follows:

with the additional notation x(t)=xc(t)+xd(t), n(t)=x(t)+y(t)+1, uc(t)=min⁡(cxc(t),μn(t)xc(t)/x(t)), and ud(t)=min⁡(cxd(t),μn(t)xd(t)/x(t)). In order to find the steady state values, the equilibrium values for the number of each class of leeches and seeds are calculated according to the same procedure described in [²]:

The Markov chain presented in the previous subsection is also extended according to the two new 2-class model. The new model is the triple (Lc(t),Ld(t),S(t)), with the obvious meaning. The initial state is (0,0,1). Starting from state (nc,nd,m), the transition rates are: λc to state (nc+1,nd,m) and λd to state (nc,nd+1,m); nθc to state (nc−1,nd,m) (for nc≥1) and nθd to state (nc,nd−1,m) (for nd≥1); (m−1)γ to state (nc,nd,m−1) (for m≥2); min⁡{cnc,μ(nc+nd+m)nc/(nc+nd)} to state (nc−1,nd,m+1) (for nc≥1) and min⁡{cnd,μ(nc+nd+m)nd/(na+nd)} to state (nc,nd−1,m+1) (for nd≥1).

3.3 P2P Network Simulation at Peer level and Constant Time Rates

In this section, the event simulation model at peer level is discussed in detail. The main objective of this simulator is to study a managed P2P network where the transfer rates are constant. Additionally, it allows to verify the results derived form the mathematical models and to obtain different performance metrics in order to have more insight of a P2P system.

As such, the simulations are performed as close as possible to both the Markov and Fluid models but without some important assumptions. Specifically, the aforementioned models consider that all the peers, from the moment of the arrival of the peer to the moment of the peer departure, always cooperate to share the file in the system. Conversely, the simulation considers that the peers can only share the file once it has downloaded the file. A more detailed simulation, considered for future work would consider the sharing of the file once it has downloaded at least some chunks of the file.

Another important difference is that, the simulator considers that whenever the active peers are uploading the file to other peers, they upload at the maximum rate μ. Hence, it cannot serve more than one peer at the time. On the other hand, the mathematical models consider that whenever a new peer arrives, the rest of the active peers in the network immediately begin to share the file, hence any given peer can serve more than one peer simultaneously.

The pseudo-code of the event simulator is presented in Algorithm 1. Four types of events are considered: Arrival of a new peer, End of the download of the file, Departure of a Leech, and Departure of a Seed. Each arriving peer is assigned an individual and particular identifier id. Hence, in the pseudo-code, the identifier of the peer that is being attended according to the different events is referred as id. And the identifier of the peer that is uploading the file is referred as idupload.

Algorithm 1 Pseudocode of the Event Simulator 

In the peer arrival event, the new peer looks for an idle peer, i.e., a peer that is not uploading the file to any other peer. Indeed, since a peer that shares the file uploads at the upload rate μ it cannot upload to any other peer. And the new arriving peer continues to look for other peers until the download capacity is fulfilled. For each peer that shares the file, it is marked as busy as an indicator that it cannot upload the file to another peer. When the file is downloaded, these uploading peers are marked as idle and they can now attend other peers. Also, the peer that finished the download is converted to a seed and is also marked as idle. Since a new peer programs its departure of the system, whenever an End Download event occurs, all the events related to the peer id have to be removed from the event list. This is also the case of the Departure Leech event. However, for the Departure Seed event, this is not necessary since a Seed only has this event on the event list. Finally, the expo instruction corresponds to the generation of an exponential distributed random variable.

An additional advantage of this event simulator is that it allows the use of different distributions for the time variables. In particular, we want to investigate the system when the transfer rates are constant. The interest of fixing the download and upload rates to constant values, is that it models one important aspect of managed P2P networks. As such, it is possible to evaluate the impact of the priority schemes on these systems. Indeed, in a managed P2P network, the service providers use their own devices which typically have the same upload and download rates. This is an important difference compared to the case of a more general P2P system, where users typically use their PCs or mobile devices (laptops, smart phones, etc.). It is clear that these devices have very different capabilities for both hardware and software. Also note that the Markov model and Fluid model are no longer valid in this case. Finally, the implementation of the constant model in the simulator is straightforward.

4.1 Priority Scheme for Cooperative Leeches

As stated before, the basic idea of the first proposed priority scheme model is to first share the file with the high tolerance leeches in case of penury conditions. It is important to notice that there can be two cases of penury. The first one occurs when the uploading bandwidth is not enough to serve at the maximum capacity all peers in the system but it is enough to do it for all thebcooperative leeches. In this case, the remaining bandwidth is shared among the defeat leeches. The second penury case corresponds to the situation where the uploading bandwidth is not enough to serve the cooperative leeches at maximum capacity. In this case, the defeat leeches are not served at all and the uploading bandwidth is only shared among cooperative leeches. In view of this and based on (4)-(6), the value of ud can be written as min⁡(cxd,max⁡(μ(x+y+1)−cxc,0)).

Let us first focus on the case of abundance. In this condition, the equilibrium values for the number of leeches and seeds in the system are the same as the equations for the two population model in case of abundance, i.e., (7), and (8) with β=c. Now, for the first case of penury where defeat leeches cannot be served at maximum capacity but there is abundance for the cooperative leeches (cx>μ(x+y)>cxc), the equilibrium point is given by the following expressions:

where

Finally, for the second penury case where there is only available bandwidth for the cooperative leeches, i.e., (cxc>μ(x+y)), the equilibrium values are calculated as follows:

4.2 Priority Scheme for Cooperative Seeds

In this second scheme, the peers that are statistically more likely to remain in the system longer once they have downloaded the file are served first over the peers that defeat once they become seeds. For this model, the number of cooperative and defeat seeds have to be considered separately. Therefore, the two population model presented before has to be extended accordingly. It is important to emphasize the difference between this priority scheme and the priority scheme for cooperative leeches. In the later, each peer is studied regarding their behavior as a leech and the leeches that are statistically more likely to rest in the system are classified as cooperative peers. These peers who have a high probability of staying in the system throughout the download procedure are served first in case of penury. For the former, each peer is studied regarding their behavior as seed and the ones that are statistically more likely to remain in the system as seeds are classified as cooperative peers.

These cooperative peers are served first in case of penury. As such, in the priority scheme for cooperative leeches, the peers are classified as having a high download aborting rate θd or a low download aborting rate θc while all seeds have the same leaving rate γ. On the other hand, for the priority scheme for cooperative seeds, the leeches are considered as having the same download aborting rate θ while the seeds are classified as having a high leaving rate γd and low leaving rate γc.

First, the two population model with no priorities is presented. The fluid model presented above is extended for the two types of peers as follows:

with uc(t) and ud(t) as defined in the Section 3.2. The equilibrium values for the number of each class of leeches and seeds in abundance conditions (cxc(t)≤μ(xc+xd+yc+yd+1)) are calculated as:

However, the case of penury (cxc(t)>μ(xc+xd+yc+yd+1)) is much more complicated. Hence the system in this condition is studied using the Markov model and through the discrete event simulator at the peer level. The Markov chain for this priority scheme for cooperative seeds is given by (Lc(t),Ld(t),Sc(t),Sd(t)) with the obvious meaning. Since in this model the interest is to analyze the performance of the system in the presence of cooperative and defeat seeds, the leeches are considered to have the same behavior. Specifically, it is considered that θc=θd=θ and γc<γd. The initial state is (0,0,1,0). Starting from state (nc,nd,mc,md), the transition rates are: λa to state (nc+1,nd,mc,md) and λd to state (nc,nd+1,mc,md); ncθ to state (nc−1,nd,mc,md) (for nc≥1) and ndθ to state (nc,nd−1,mc,md) (for nd≥1); (mc−1)γc (nc,nd,mc−1,md) (for mc≥2) and mdγd to state (nc,nd,mc,md−1) (for md≥2); min⁡{cnc,μ(nc+nd+mc+md)nc/(nc+nd)} to state (nc−1,nd,mc+1,md) (for nc≥1) and {cnd,μ(nc+nd+mc+md)nd/(nc+nd)} to state (nc,nd−1,ma,md+1) (for nd≥1).

The fluid model with priorities is basically the same as the model with no priorities except for the rates uc and ud. These rates are modified in a similar manner as in the priority model for cooperative leeches and are now described by: uc=min⁡(cxc,μ(x+y+1)) and ud=min⁡(cxd,max⁡(μ(x+y+1)−cxc,0)). It is worth noting that when the system is in abundance conditions, the equilibrium values for the number of leeches and seeds are also described by (19)-(20). For the case of penury for all peers in the network, the equilibrium values are described by:

The system in penury only for the defeat seeds is studied through the Markov model and the discrete event simulator due to its complexity. Similarly, the Markov chain is only modified in the following transition rates: min⁡{cnc,μ(nc+nd+mc+md)} to state (nc−1,nd,mc+1,md) (for nc≥1) and min⁡{cnd,max⁡(μ(nc+nd+mc+md)−cxc,0)} to state (nc,nd−1,ma,md+1) (for nd≥1).

5.1 Exponential Distributed Download and Upload Rates

In this subsection, the exponential distributed model for the transfer rates in the simulation at peer level and mathematical models are considered. Recall that the simulation under the exponential model is constructed in order to recreate as close as possible the scenario considered by the Markov and Fluid model presented above, i.e, the download and upload rates are modeled as exponentially distributed random variables with mean c and μ respectively.

In Fig. 2, the evolution in time of the number of leeches and seeds normalized by the arrival rate is presented. It can be seen that there is a very good match between the simulation at peer level results compared to the fluid model, specially for high values of λ for both the number of seeds and leeches.

Fig. 2 Evolution in time of the number of Leeches and Seeds in the network, Simulation at peer level vs Fluid Model 

In Fig. 3, the average download (time since the arrival of the peer to the instant that the complete file is downloaded), waiting (time form the arrival of the peer to the moment of the leech departure before downloading the complete file) and life time (time form the arrival of the peer to the moment when the peer either completes de download of the file or leaves the system aborting the download) as leech and the proportion of successful downloads is presented. Various interesting observations can be made from these results.

Fig. 3 Average Download, Waiting, Life Times for Leeches (sec) and Proportion of Successful Downloads 

First of all, it can be seen that the analytical average times derived analytically (under the fluid model), are always the same, independent of the arrival rate. The reason for this can be explained as follows: Any leech can either leave the system either because it abandons the download, with rate θ or because it finishes the download and becomes a seed with rate c (for simplicity consider abundance conditions, i.e., cx<μ(x+y))). Notice that both of these times, i.e. the time that the leech stays in the system before quitting the download and the time that a leech stays in the system before becoming a seed, are independent exponentially distributed random variables. It is well known then, that the minimum of these times can be modeled by a third random variable with rate c+θ. Hence, the average time that a peer stays in the system as leech, either because it downloads the file or because it aborts the download is always 1/(c+θ), which is the value presented in the three graphics for the fluid model.

A second observation is that the simulation results show a slightly higher average times compared to the analytical results. The reason for this is that, in the analytical model, whenever a new peer arrives, it is assumed that it immediately starts to share the file with other peers and at the same time it begins to download the file. On the other hand, the simulation considers that whenever a new peer arrives, it can not upload the file to other peers since it has not yet downloaded the file. And also, if the rest of the peers are already uploading to other peers and there is not an idle peer, it can not start the download of the file. Hence, the download time is slightly higher. And because leeches spend more time in the network before becoming seeds, there is a higher chance of quitting the download, therefore the waiting time is also incremented and consequently the average life time as leech also increases compared to the analytical model. These observations are further reaffirmed by observing the probability that a leech successfully downloads the file in Fig. 3 (d). In the analytical model the probability that a leech becomes a seed is simply c/(c+θ). In the simulations however, this probability is slightly lower.

5.2 Fixed Download and Upload Rates for Single Population Model

In this subsection, the results when the transfer rates are constant instead of random variables are presented and compared to the exponential model. In Fig. 4, the number of leeches and seeds normalized to λ are presented. For reasons of clarity, only the results for λ=4 are shown. The case where μ and/or c constant are studied. Recall that the exponential model for the transfer rates is not well suited due to its high variability which is not expected to happen in a practical case, specially on a managed P2P network.

Fig. 4 Evolution in time of the number of Leeches and Seeds in the network with λ=4  

It can be seen that by leaving any of the transfer rates constant, the number of leeches is higher than the exponential model and consequently, the number of seeds is smaller. In particular, when both μ and c are fixed, the greatest difference can be appreciated compared to the exponential model. This observation is further confirmed in Fig. 5, where the average number of seeds and leeches is presented. It can be seen that the constant model for both transfer rates also shows the greatest difference for any value of λ. This implies that when both transfer rates are fixed, the average download time is the highest as depicted in Fig. 7(a).

Fig. 5 Average Number of Leeches and Seeds for c and/or μ Constant 

Fig. 6 Variance of the Number of Leeches and Seeds for c and/or μ Constant 

Fig. 7 Average Download, Waiting, Life Time of Leeches and Proportion of Successful Downloads 

Indeed, it is clear form these results that the exponential model allows a fastest download time and corresponds to an optimistic model for a P2P network. Hence, the average time that a peer stays as leech in the system is also the highest when the transfer rates are fixed as observed in Fig. 7(c) and also, the successful download percentage is the lowest for the same case observed as seen in Fig. 7 (d).

Another interesting observation is that for the case where μ is fixed while c is variable, the variance in terms of the number of leeches and seeds is the highest. This is a good indication that a small variation in the mean value of μ has a big impact on the randomness of the system. However, the overall performance of the system is not considerable different in terms of the download times and probability of successful download.

5.3 Two Population and Priority Models for Exponential Transfer Rates

In this section, the performance of the proposed priority mechanisms are evaluated.

5.3.1 Priority Model for Cooperative Leeches

For these results, different portions of high tolerance leeches are considered, ranging from 1% through 90% of the total leech population. Also, two different system conditions are considered: abundance, where the number of seeds in the system is high enough as to have most of the time all leeches downloading at the maximum capacity. As such, a value of γ=0.001, is considered. The second case is penury, where the number of seeds in the system is very low. Hence the uploading bandwidth is very scarce. To obtain this condition, a value of γ=0.005 is considered.

The case of abundance can be seen as a case where most peers remain in the system even after the download of the file in order to cooperate with the file download. And the case of penury can be seen as the case where once the file is downloaded, the peers almost immediately leave the system. It is important to keep in mind that seeds’ behavior analysis is a different issue than the one investigated in this work. Recall that we are focusing on the behavior of the leeches while they are downloading the file, i.e., if they have a high probability of aborting the download or not. Nevertheless, this priority scheme can also be implemented focusing on the behavior of the seeds, which we plan to study in a future work.

Another important remark about the results presented in this section is that, as stated in [²], the fluid model is more accurate when the arrival rate λ is high. Hence, we use the fluid model for λ=4, while the Markov model is used to obtain results for λ=0.04, 0.4; in order to verify the analytical results, the value of λ=4 is also considered.

In Fig. 8, the normalized equilibrium values for the number of leeches and peers under the proposed priority scheme is presented using the fluid model when λ=4 and for different proportions p of high tolerance leeches in abundance conditions.

Fig. 8 Normalized Equilibrium Number of Leeches and Seeds for Different Portions of High Tolerance Leeches, Fluid Model, Abundance 

The normalization is made with respect to λ. In Fig. 9 we present the same results but using the Markov model and for λ=0.04, 0.4 and 4. First, it is important to remark that both models present a very close mach, validating somehow each other. Also, under this condition of abundance, the values for the normalized number of leeches and seeds is exactly the same for the system with and without priorities as described in previous section.

Fig. 9 Normalized Equilibrium Number of Leeches and Seeds for Different Portions of High Tolerance Leeches, Markov Model, Abundance 

Indeed, the priority scheme has no impact on the system when the uploading bandwidth is high enough as to allow all leeches to download at the maximum capacity c.

In Fig. 10, the probability that a leech successfully downloads the file is presented. Again, there is no difference between the conventional system and the system with the priority scheme enabled. It is important to notice that as the number of high tolerance leeches increases, the probability of successful download also increases.

Fig. 10 Probability of Total Successful Downloading, Abundance 

This is because high tolerance leeches remain longer in the system and have a high probability of remaining throughout the whole downloading process. Conversely, when the fraction of high tolerance leeches is low (for example p=0.01), there is a high probability that the downloading leeches abort the download and leave the system.

In penury conditions, the priority mechanism has a visible impact on the system since, in that case, high tolerance leeches are served first. For instance, Fig. 11 shows the normalized equilibrium value of the number of leeches and seeds for λ=4 derived from the fluid model. It can be seen that when the priority mechanism is enabled, the normalized number of high tolerance leeches is higher than for the conventional system. More importantly however is the fact that the normalized number of seeds when the priority mechanism is enabled is higher than for the conventional case.

Fig. 11 Normalized Equilibrium Number of Leeches and Seeds for Different Portions of High Tolerance Leeches, Fluid Model, Penury 

This effect is more visible when there is a high portion of high tolerance leeches in the system, since in this case there is more cooperation among peers. Therefore, there are more leeches that download the file and become seeds. This is “paid” by the low tolerance leeches that are not being served and leave the system before having a chance of downloading the file.

This observation is reinforced by observing the results in Fig. 13 (a) which shows the probability of successfully downloading the file. It can be seen that for different values of λ this probability is always higher for the high tolerance leeches when the priority scheme is enabled. In fact, for high values of λ and high percentage of high tolerance leeches in the system, whenever a high tolerance leech initiates the download of the file, it will successfully download it. In other words, for these leeches, the probability to successfully download the file is (very close to) one.

Fig. 12 Normalized Equilibrium Number of Leeches and Seeds for Different Portions of High Tolerance Leeches, Markov Model, Penury 

Fig. 13 Successful download in penury 

Conversely, the probability that a low tolerance leech downloads a file is almost zero. Recall that low tolerance leeches are not rewarded, but notice that not rewarding the low tolerance leeches is different from being punished. The difference is that the peers are not wasting resources in punishing these leeches, they are just ignoring them whenever the system is in penury conditions. Since the low tolerance leeches are not likely to remain in the system throughout the complete download process, the global resources are better used by serving first the more tolerant users. Also, the less tolerant leeches do not get served unless there are enough resources in the system, and they are more likely to abort the download reducing significantly the successful download probability.

Notice that the results from the Fluid Model in Fig. 11 are validated by the results from the Markov model presented in Fig. 12. Observe that there is a better fit between these two models when λ is high, as already mentioned in [²].

Finally, the total successful download probability is presented in Fig. 13 (b). In this case, the percentage of leeches (both high and low tolerance) is lower when the priority scheme is enabled compared to the conventional system. The reason behind this is that, as already stated, the high tolerance leeches are rewarded by serving them before the low tolerance ones.

Hence, these low tolerance leeches, which already have a low latency tolerance, are more prone to leave the system. That is, for the conventional system with the priority scheme disabled, the probability that some low tolerance leeches download the file is not zero because the peers cooperate with any leech independently of its tolerance to the download delay. As such, the probability that a leech in general downloads the file is higher when no priority scheme is being used.

5.3.2 Priority Model for Cooperative Seeds

In this section, the case where the cooperative seeds are served first is discussed in detail. For these experiments, the system parameters are: c=0.002, μ=0.0012, θ=0.001, λc=pλ, λd=(1−p)λ, γ=0.001, 00.005, γc=γ, and γd=5γ. We highlight the main difference between these two models. In the Markov model, the cooperative peers use the available bandwidth among each other while the defeat peers can only use the remaining bandwidth. However, for the simulation model, not all peers can cooperate among each other. Recall that arriving peers look for available peers until they achieve the maximum download rate or no active peer can download him. Hence the case where no available peers are found can occur. For the priority schemes, when a cooperative peer arrives, it looks for peers to download the file and if it does not found one, it looks for a peer that is serving a defeat peer. If there is one defeat peer being served the downloading peer stops serving the defeat leech and starts to serve the cooperative leech. This cooperative peer continues this procedure until it achieves the maximum downloading rate or no more peers are available. This issue provides important differences among the results of the two models.

Let us first discuss the Markov model. In Fig. 14 the normalized (on λ) number of peers for the priority scheme that favors the cooperative sseds when the system is stable for γ=0.001 is presented while in Fig. 16 the same results are presented for γ=0.005. As such, Fig. 16 corresponds to a harsher environment compared to the case presented in Fig. 14, since seeds remain shorter times in the system. In Fig. 15, the total download probability is depicted for γ=0.001. It is worth mentioning that for low concentrations of cooperative seeds, the probability of downloading the file is slightly lower than for the case of high concentration of cooperative peers. In other words, this probability increases as the concentration of cooperative peers increases until a point (close to 20%) where it stabilizes. The reason for this is that, when γ=0.001 the system has enough seeds (even if many of them are non cooperative) to serve almost any leech arriving to the system. Hence, most of the peers that do not abort the download become seeds. As such, any increment on p after 0.2 has no visible effect on the probability of downloading the file. On the other hand, observe Fig. 17, where γ is much higher (i.e., seeds remain in the system for shorter times), there are not enough resources in the system to serve arriving leeches. Therefore, any increase on the concentration of cooperative seeds, increases the probability of successful download. In this sense, it is fair to say that for γ=0.01 the system is in abundance for p≥0.2, while for γ=0.005, the system is in penury for any value of p. Comparing the performance of the system under the priority scheme that benefits the cooperative seeds with the scheme that favors the cooperative leeches, observe Figs. 10, 13 (b), Fig. 15, and Fig. 17. It is clear that the priority scheme that favors cooperative seeds has a better performance in terms of total download probability than the priority scheme that favors the cooperative leeches. Also note that, when the cooperative seeds are prioritized, in case of penury, the probability of successful download is higher than the case where no priorities are enabled. Recall that the priority scheme that favors cooperative leeches, in the same conditions of penury, the probability of successful download is higher when no priorities are enabled.

Fig. 14 Number of Peers for Different Portions of Cooperative Peers, Priorities for Seeds, γ=0.001, Markov Model 

Fig. 15 Proportion of Total Downloads, γ=0.001, Markov Model 

Fig. 16 Number of Peers for Different Portions of Cooperative Peers, Priorities for Seeds, γ=0.005, Markov Model 

Fig. 17 Proportion of Total Downloads, γ=0.005, Markov Model 

We now consider the simulation’s results. The performance of the system under the two priority schemes is presented For γ=0.001, Fig. 18 presents the average number of peers in the system for the priority scheme for cooperative leeches while Fig. 19 presents the results for the priority scheme for cooperative seeds. It is clear that for the first priority scheme, there is only one class of seeds. Hence, only the total number of seeds is shown.

Fig. 18 Number of Peers for the Priority for Cooperative Leeches Scheme, Simulation, γ=0.001  

Fig. 19 Number of Peers for the Priority for Cooperative Seeds Schemes, Simulation, γ=0.001  

On the other hand, for the second priority scheme, the two classes of seeds is shown. From these results it is clear that as the number of cooperative peers increases, also the total number of seeds increases. This is because the peers rest in the system longer times either as leeches, completing a higher number of downloads, or as seeds, remaining longer times after the download.

In Fig. 20, the average download time for the two priority schemes and the system with no priorities is presented. for reasons of clarity, only the results for λ=0.4 and λ=4 are shown. In Fig. 20 (a) the performance of the system for the cooperative peers is presented. It can be seen that the lowest download times corresponds to the scheme where the cooperative seeds are benefited. As expected, when the peers that are statistically more likely to remain longer in the system after the download are served first, there is a higher probability of finding a peer with the complete file and hence, the file is downloaded faster.

Fig. 20 Average Download Time for Priority Schemes, Simulation, γ=0.001  

For the priority scheme that favors the coopera-tive leeches, the download time is higher because it only benefits the peers that stay longer during the download procedure. However, these peers may quit the system almost immediately after downloading the file. As such, it is more profitable to benefit the peers that stay longer sharing their resources once they complete their download than favoring those who are more likely to stay throughout the download but may act selfishly afterwards.

Also, when no priorities are enabled in the system, the average download time is the highest. The highest download time for the defeat peers corresponds to the priority scheme that favors the cooperative seeds. Again, this is due to the fact that defeat peers are only served after the cooperative peers are served. Hence it takes more time for them to download the file. One important issue is the total download time in the system.

It can be seen that the priority scheme for the cooperative leeches achieves the lowest download times for all peers in the system while the priority scheme for cooperative seeds achieves a low download time only after a certain number of cooperative seeds are in the system.

The rationale behind this can be explained as follows: when the cooperative seeds are favored and the number of cooperative seeds is low, the system serves first these few peers which does not have a great impact on the system since most of the seeds quit the system very fast after their download.

Hence, the download time for the rest of the peers (which are defeat seeds) is high. As the proportion of cooperative peers increases, the probability of having cooperative seeds in the system increases, reducing also the download time. For the case when the cooperative leeches are favored, when the number of cooperative peers are in the system the download time is low since the model considers that all seeds remain in average the same amount of time (1/γ), therefore, the number of seeds is high compared to the second priority scheme (where most seeds remain only 1/γd which is a much shorter time).

In Fig. 21, the proportion of successful downloads for the two priority schemes and the total number of downloads is presented. In the case of the cooperative peers, the scheme that achieves the best results is the one that favors the cooperative leeches, as it is shown in Fig. 21 (a). Conversely, for the defeat peers the priority scheme that benefits the cooperative peers achieves the highest proportion of successful downloads. Also, this scheme obtains the highest total proportion of successful downloads for all peers in the system. Indeed, as the system serves first the peers that as seeds stay longer in the system, all peers benefit from having a higher probability of finding a peer with the complete file willing to cooperate. As such, the proportion of total successful downloads is higher.

Fig. 21 Proportion of Downloads for the Priority Schemes, Simulation, γ=0.001  

On the other hand, for the cooperative peers, the scheme that benefits the seeds that remain longer has the lowest proportion of downloads specially for low concentration of cooperative peers. This is because there is a high number of non cooperative peers which leave the system almost immediately after downloading the file. In the same conditions, the priority scheme that benefits the cooperative leeches, considers that all seeds have the same rate γ, which implies a higher number of seeds. This is turn increases the number of downloads since the leeches benefit from a high number of seeds irrespectively of the concentration of cooperative leeches.

5.4 Two Population and Priority Models with Constant Transfer Rates

In this section, the effect of the constant transfer rates are discussed in detail. The cases where the download and/or the upload rates are constant are considered. For these experiments the system parameters are as follows: λ=0.4, θ=0.01, c=0.002, μ=0.00125, γ=0.01 for abundance conditions and γ=0.05 for penury conditions. First, we focus on the average download time.

It can be seen in Fig. 22 that in the system where c and/or μ are constant the average download time increases compared to the exponential model under abundance conditions. This implies that the high variability on the transfer rates implicit on the exponential model corresponds to a optimistic system in the sense that the system is benefited by allowing both very high and very low values for these transfer rates, which is not expected on a managed P2P network.

Fig. 22 Average Download Time for Priority Schemes, Abundance 

Also, from these results, it can be seen that by considering only c fixed and μ exponentially distributed, the system behaves in the same manner as the only exponential model, i.e., the system whit the priority scheme for cooperative leeches and the system with no priorities start form a low value of download time and increases with p while the system under the priority scheme for cooperative seeds starts form a high value of download time and decreases for p.

Conversely, when μ or both transfer rates are constant, the behavior of the system is the opposite. As such, it is clear that the system is very sensitive on the model used for the parameter μ The same observations can be drawn for the system under penury conditions as it is shown in Fig. 23.

Fig. 23 Average Download Time for Priority Schemes, Penury 

Therefore, the impact of the model of the transfer rates is independent on the system’s conditions. It is worth noting that the priority schemes are no longer effective when μ and/or both transfer rates are constant, since for high concentrations of cooperative peers, the system with no priorities has the best performance.

Similarly, for the probability of successful download, a similar effect is presented as shown in Figs. 24 and 25. First, this probability is lower for the all exponential case. Second, the case where only c is constant provides a similar behavior than the exponential model where the system with no priorities has the worst performance for any value of p while for the models where μ and/or both transfer rates are constant for high concentration of cooperative peers, the system with no priorities obtains the best results, i.e., the behavior of the system drastically changes.

Fig. 24 Proportion of Total Downloads for Priority Schemes, Abundance 

Fig. 25 Proportion of Total Downloads for Priority Schemes, Penury 

Again, we conclude that the parameter μ is highly sensitive to the selected model. It is worth noting that, for any model used, and for low concentrations of cooperative peers, the priority scheme that favors the cooperative seeds, obtain a much higher probability of successful download. These are encouraging results since this priority scheme can be used for both managed or unmanaged P2P networks in order to increase the successful download probability in particular for a system where a high portion of seed behave in a selfish manner.