“…Let p ww , p wc , p cw , and p cc be the probabilities of a state request status, r ∈ R = {0, 1, 2}, transitioning in the SMDP as 0→0, 0→{1, 2}, {1, 2}→0, and {1, 2}→{1, 2}, respectively. Then, p ww = e −λ∆0 (if no request is received, the SMDP remains in the waiting state), p wc = 1 − p ww , p cw = 1, and p cc = 0 (if the SMDP is in the communication state, the next state of the SMDP will be a waiting state, see (9)). Therefore, the steady-state probabilities of being in the waiting and communication states, π wait and π comm , satisfy π wait = p ww π wait + p cw π comm = e −λ∆0 π wait + π comm , π comm = p wc π wait + p cc π comm = (1 − e −λ∆0 )π wait , π wait + π comm = 1, whose solution is given as in the statement of the lemma.…”