###### Abstract
Additionally, there is a transitive property, i.e., if node A and node B contact frequently, and if node B and node C contact frequently, it is highly probable that node A and node C contact frequently, as defined in Equation (3) [8]:  
\[\rm P(A,C)=MAX\left(P(A,C)old,\ P(A,B)*P(B,C)recv\times\beta\right), \tag{3}\]  
where \(\beta\) is a scaling constant. In the PRoPHET protocol, when two nodes contact, i.e., when they are within the transmission range, they exchange summary vectors which include all the delivery predictability values of all previous contact nodes [8]. In this paper, the PRoPHET protocol is considered as a base protocol. In PRoPHET, several forwarding strategies are defined. In the GRTR strategy, for example, a message to a destination node to D is forwarded to node B from node A, if P(B,D) is larger than P(A,D). In the GTMX strategy, a message is forwarded if P(B,D) is larger than P(A,D) and the number of forwarding of the message considered is smaller than a threshold value NFmax. In GRTR+, a message is forwarded to if P(B,D) is larger than P(A,D) and P(B,D) is larger than Pmax, which is the largest delivery predictability stored by a node to which the message has been forwarded to ever. We note that GRTR, GTMX, and GRTR+ are just mnemonics [8]. In this paper, we propose an efficient opportunistic routing protocol using the delivery predictability of previously contacted node, i.e., the history of delivery predictability. In the proposed protocol, if a node receives a message from another node, the current value of the delivery predictability of the receiving node to the destination node of the received message is stored as previous the delivery predictability (preP) in the receiving node. We note that the value of preP of a node is defined for each destination node individually and the value of preP for a node is updated whenever it receives a message destined to the node. When two nodes contact, a message is forwarded to the other node, if the delivery predictability of the other node is higher than both the delivery predictability and preP of the sending node. If there is no value of preP for the considered message, only the current delivery predictability values are compared and a message is delivered to the other node if the delivery predictability of the other node is higher. The preP can be considered as an indicator of a good message forwarder based on the history of delivery predictability. That is, if a node A has a higher preP than the current delivery predictability of the contact node B, although node A has a smaller delivery predictability currently than that of the contact node B, it is not appropriate to disseminate the message to node B. This is because too much forwarding based on only current delivery predictability result in the message dropping in the buffer due to buffer overflow and, thus, the overall performance degrades. In the PRoPHET protocol with the GRTR strategy, if a node with a message has a small delivery predictability, the message is forwarded significantly and results in the message dropping in the buffer. In order to restrict the message dissemination efficiently in the PRoPHET protocol, enhanced message forwarding strategies such as GTMX and GRTR+ were proposed. In the PRoPHET protocol with the GTMX strategy, the total number of forwarding of the message is limited by a threshold value in addition to the basic GRTR strategy comparing current delivery predictability. In the PRoPHET protocol with the GRTR+ strategy, the additional condition is that the delivery predictability of the receiving node is larger than a threshold probability, which is the largest delivery predictability reported by a node to which the message has been sent so far. In the PRoPHET protocol with the GRTR+ strategy, the threshold probability is a non-decreasing function of time and message forwarding is too restricted as time goes on, which may result in an overall performance degradation.