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A wide range of emergency vehicle location models have been proposed,22 23 all with their different pros and cons, and future research should include method comparison studies of various modelling approaches. The mathematical model used in the present study is an idealised version of the problem under study, yet one with significant practical relevance. The model assumes that whenever there is need for a vehicle at a base station, there is always one available. In this sense, the model represents a best-case scenario. If a geographical location cannot be reached within the specified target time in the MCLP model, it never can. In practice, the assumption that an ambulance is always available at a base station whenever needed will not necessarily hold true, in particular in busy areas, that is, places with frequent injuries. Including a parameter for such a busy fraction could change the base locations.
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In this work, we focus on file distributing networks, and on the harmful effect caused by peers that enter the system and begin the file downloading process but leave before finishing it or just after finishing their own download. Hence, resources can be assigned to a peer that is not staying long enough in the network to cooperate with the rest of the users, that is, leading to a waste.
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).
We built our priority model based on , and using also [3, 4, 13]. Unlike these works, we consider a two-population model. Additionally, a priority mechanism is proposed and studied by means of a fluid simplification and a Markov chain. Fluid models as the one in , allow simple analytical discussions. We complete the analysis with a Markov chain model such as the one presented in  to get better insight into the performance of the system. In , two classes of users are considered, namely: high bandwidth users and low bandwidth users. This is done in order to approximate a real system where different users have different hardware characteristics. Unlike , we focus on the behavior of the different peers during the downloading procedure rather than the different bandwidth capacities that can be encountered in a real network. Moreover, an event simulator is implemented to study a managed P2P network where the transfer rates are considered to be constant. This model can no longer be studied using neither the Markov chain nor the Fluid model.
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 , 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:
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.
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.
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