July 16, 2017

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OT Þ can be observed. 4 The observed workload data of a Web site consists of ot, the number of user requests per second, with the maximum observed value K 5 max{ot} 5 74 requests/s. The total number of observations is T 5 3600 s (over 1 h) in the workload data set. The user request arrivals are governed by an underlying hidden semi-Markov process. The hidden state represents the arrival rate, which is corresponding to the number of users that are browsing the website. 7 Data (requests/s) and the hidden states of the workload.

If the simplified assumption that the first state must start at t 5 1 is assumed, let ατ ðj; dÞ 5 0 for τ , 0; otherwise let ατ ðj; dÞ 5 πj;d for τ , 0; 3. For t 5 1; . ; T { for j 5 1; . ; M and d 5 1; . ; D { αt ðj; dÞ 5 X at2d ði; hÞaði;hÞðj;dÞ bj;d ðot2d11:t Þ; i;h } } The Backward Algorithm 1. For j 5 1; . ; M and d 5 1; . ; D, let β T ðj; dÞ 5 1; 2. If the simplified assumption that the last state must end at t 5 T is assumed, let β τ ðj; dÞ 5 0 for τ . T; otherwise, let β τ ðj; dÞ 5 1 for τ .

In this case, we can assume that the process starts at 2N and terminates at 1N. The observations out of the sampling period [1,T] can be any possible values, that is, bj;d ðUÞ 5 1 General Hidden Semi-Markov Model 35 for any jAS; dAD. 7) bi;h ðot11:t1h Þ is replaced with bi;h ðot11:T Þ if t 1 1 # T and t 1 h $ T. 8) where fπj;d g can be the equilibrium distribution of the underlying semi-Markov process. 7) we can see that β t1h ði; hÞ 5 1, for t 1 h $ T. 9) If the model assumes that the first state begins at t 5 1 and the last state ends at or after observation oT , it is a right-censored HSMM introduced by Guedon (2003).

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