Fractional Speech Loss and Talker Activity Model for TASI and for Packet-Switched Speech

The cutout fraction in a TASI system is shown to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ=1/np∑k=c+1^n (k-c) n k pᵏ(1-p)ⁿ⁻ᵏ</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is the number of sources, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> is the number of channels, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> is the probability that a source is issuing a talkspurt at a random time. This result is shown to hold independently of the probability density function of talkspurt duration. The same formula is shown to apply to the fraction of packets lost in a packet-switched link with a transmission capacity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> packets every T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> seconds, where T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> is the interval between packet generations for an individual source during talkspurt, and where no packet is queued for a time longer than T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> . In addition, a simple Markov birth-death model is presented for the random process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(t)</tex> representing the number of talkers issuing talkspurts at a given time.