Sec.4.2 Slotted Multiaccess and the Aloha System 275 4.1.4 Packet Radio Networks A fourth example of multiaccess communication is that of a packet radio network.Here each node is in reception range of some subset of other nodes.Thus,if only one node in the subset is transmitting,the given node can receive the transmission,whereas if multiple nodes are transmitting simultaneously in the same band,the reception will be garbled.Similarly,what one node transmits will be heard by a subset of the other nodes. In general,because of different noise conditions at the nodes,the subset of nodes from which a given node can receive is different from the subset to which the given node can transmit.The fact that each receiver hears a subset of transmitters rather than all transmitters makes packet radio far more complex than the other examples discussed.In the next four sections,we study multiaccess channels in which a receiver can hear all transmitters,and then in Section 4.6.we discuss packet radio networks in more detail. 4.2 SLOTTED MULTIACCESS AND THE ALOHA SYSTEM Satellite channels,multidrop telephone lines,and multitap bus systems all share the feature of a set of nodes sharing a communication channel.If two or more nodes transmit simultaneously,the reception is garbled,and if none transmit,the channel is unused.The problem is somehow to coordinate the use of the channel so that exactly one node is transmitting for an appreciable fraction of the time.We start by looking at a highly idealized model.We shall see later that multiaccess channels can often be used in practice with much higher utilization than is possible in this idealized model,but we shall also see that these practical extensions can be understood more clearly in terms of our idealization. 4.2.1 Idealized Slotted Multiaccess Model The idealized model to be developed allows us to focus on the problem of dealing with the contention that occurs when multiple nodes attempt to use the channel simultaneously. Conceptually.we view the system as in Fig.4.1(a).with i transmitting nodes and one receiver. It will be observed that aside from some questions of propagation delay.this same model can be applied with m nodes that can all hear each other (i.e.,the situation with a multitap bus).We first list the assumptions of the model and then discuss their implications. 1.Slotted svstemt.Assume that all transmitted packets have the same length and that each packet requires one time unit (called a slot)for transmission.All transmitters are synchronized so that the reception of each packet starts at an integer time and ends before the next integer time. 2.Poisson arrivals.Assume that packets arrive for transmission at each of the m transmitting nodes according to independent Poisson processes.Let A be the overall arrival rate to the system,and let A/m be the arrival rate at each transmitting node.Sec. 42 Slotted Multiaccess and the Aloha System 275 4.1.4 Packet Radio Networks A fourth example of multiaccess communication is that of a packet radio network. Here each node is in reception range of some subset of other nodes. Thus. if only one node in the subset is transmitting, the given node can receive the transmission. whereas if multiple nodes are transmitting simultaneously in the same band, the reception will be garbled. Similarly. what one node transmits will be heard by a subset of the other nodes. In general, because of different noise conditions at the nodes, the subset of nodes from which a given node can receive is different from the subset to which the given node can transmit. The fact that each receiver hears a subset of transmitters rather than all transmitters makes packet radio far more complex than the other examples discussed. In the next four sections, we study multiaccess channels in which a receiver can hear all transmitters, and then in Section 4.6, we discuss packet radio networks in more detail. 4.2 SLOTTED MULTIACCESS AND THE ALOHA SYSTEM Satellite channels, multidrop telephone lines, and multitap bus systems all share the feature of a set of nodes sharing a communication channel. If two or more nodes transmit simultaneously, the reception is garbled, and if none transmit, the channel is unused. The problem is somehow to coordinate the use of the channel so that exactly one node is transmitting for an appreciable fraction of the time. We start by looking at a highly idealized model. We shall see later that multiaccess channels can often be used in practice with much higher utilization than is possible in this idealized model, but we shall also see that these practical extensions can be understood more clearly in terms of our idealization. 4.2.1 Idealized Slotted Multiaccess Model The idealized model to be developed allows us to focus on the problem of dealing with the contention that occurs when multiple nodes attempt to use the channel simultaneously. Conceptually. we view the system as in Fig. 4.1 (a), with III transmitting nodes and one receiver. It will be observed that aside from some questions of propagation delay. this same model can be applied with III nodes that can all hear each other (i.e., the situation with a multitap bus). We first list the assumptions of the model and then discuss their implications. 1. Slotted system. Assume that all transmitted packets have the same length and that each packet requires one time unit (called a slot) for transmission. All transmitters are synchronized so that the reception of each packet starts at an integer time and ends before the next integer time. 2. Poisson arrimls. Assume that packets arrive for transmission at each of the Tn transmitting nodes according to independent Poisson processes. Let A be the overall arrival rate to the system. and let AIm be the arrival rate at each transmitting node