Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 145Completing the squares for |()| ̃Z k in Equation 3.A.6 results in ∆J ≤ 0 aslong as the conditions in Equation 3.24 through Equation 3.27 are satisfiedand with the upper bound on the buffer-length error by|( ek)|>1⎛ ⎛2 2 ⎛ Γγkvmax+ kvmax+ +2⎜γ ρ Z1−σ k( 2 − Γ)vmax⎝⎜⎝⎜ ⎝( )⎞⎞ ⎞1 k⎠⎟ ( − σ ) ⎟⎠ ⎠⎟2 2Mvmax(3.A.9)On the other hand, completing the squares for |( ek)|in Equation 3.A.6results in ∆J ≤ 0 as long as the conditions Equation 3.24 through Equation3.27 are satisfied and( )1|()| ̃Z k