Bistability in Communication Networks

Advances in the technology of modern telecommunications networks have led to considerable interest in schemes which can dynamically control the routing of calls within a network. The aim of such schemes is to adjust routing patterns within the network in accordance with varying and uncertain offered traffics, to make better use of spare capacity, and to provide extra flexibility and robustness to respond to failures or overloads. However, unless care is taken a dynamic routing strategy that appears to be beneficial may under some circumstances be detrimental. One of the simplest dynamic routing strategies is Random Alternative Routing, which operates as follows. Every call type that can arrive at the network has a fixed first choice route and a set of possible second choice routes. If possible a call will be carried on its first choice route. If not, then an alternative route is selected at random from the set of possible second choice routes. The call is carried on this route if possible and otherwise it is lost. At first sight the existence of second choice routes appears beneficial, as it gives each call more ways of being accepted. However if second choice routes require more network resource (hold more circuits or possibly hold circuits for longer) then the network performance, as measured for example by its overall loss probability, may be worse than if a call has access to just its first choice route. Further, Random Alternative Routing (and several other dynamic routing schemes) can lead to instability and hysteresis: several modes of behaviour are possible, the initial conditions of the network determining which is obtained. In this paper we use a combination of analytical, numerical and simulation approaches to investigate these phenomena, in the simplest case of a symmetric fully connected network. The possibility of bistable behaviour was first noted by Nakagome and Mori (1973), using a simple analytical fixed point approximation for the equilibrium behaviour of a network. Using a development of this approximation Krupp (1982) showed that a network’s performance could be improved by using a simple priority technique, known as trunk reservation.