Blocking Probabilities for a Single Link with Trunk Reservation

A single link in a circuit-switched network is considered. The link has C circuits, R of which are reserved for the primary directly offered.traffic. Offered calls arrive in independent Poisson streams with mean rates l and n for the primary and secondary rerouted.traffic, respectively, and corresponding independent and exponentially distributed holding times with means 1 and 1rk. A primary call requires just 1 circuit, whereas a secondary call requires t circuits, where t is a positive integer. A primary call is blocked on arrival if all C circuits are busy, whereas a secondary call is blocked if more than CyRyt circuits are busy. Blocked calls are lost to the link. The critically loaded case in which l c1, CylsO ʹl., RsO ʹl., and n sgʹl, where g sO 1., is investigated. Asymptotic approximations to B1 and B2, the blocking probabilities for the primary and secondary traffic, are derived. The results are explicit if k s1, but involve expansions in powers of g for k /1, in which it is shown how to determine the coefficients recursively. The first two terms in powers of g are given explicitly. An alternate approach, which involves truncation rather than power series expan- sions, is presented. The case RsO 1. is also considered, and explicit results are obtained. Finally, an approximation proposed by J. W. Roberts is investigated. The approximation is shown to be asymptotically correct if k s1, or if RsO 1., but not if k /1 and RsO ʹl.. Interestingly, Roberts’ approximation corresponds to truncation with just 2 coefficients. Truncation with more coefficients leads to refinements of his approximation.