Design and evaluation of overloaded service systems with skill based routing, under FCFS policies

A skill based parallel service system under FCFS-ALIS — steady state, overloads, and abandonments

“…= ∞, which has slope 1 for 0 < λ < λ (1) , and is constant and equal to µ for λ > λ (L) . (ii) For each interval (λ (i−1) , λ (i) ) there exist a set of customer types C(λ (i−1) , λ (i) ) and a set of servers S(λ (i−1) , λ (i) ) such that they form a cut, which is the unique minimal cut for all λ in the interval.…”

“…As λ increases, the subset of customers, which have the least value of β S(C) /α C , becomes overloaded when λ reaches µ β S(C) α C , and this defines C (1) and S (1) . This leaves servers S\S (1) to serve the remaining customer types C\C (1) .…”

“…As λ increases, the subset of customers, which have the least value of β S(C) /α C , becomes overloaded when λ reaches µ β S(C) α C , and this defines C (1) and S (1) . This leaves servers S\S (1) to serve the remaining customer types C\C (1) . Note that even if c ∈ C\C (1) can be served by a server in S (1) , this will not happen in the max flow solution, since all the servers in S (1) are fully occupied by C (1) .…”

“…This leaves servers S\S (1) to serve the remaining customer types C\C (1) . Note that even if c ∈ C\C (1) can be served by a server in S (1) , this will not happen in the max flow solution, since all the servers in S (1) are fully occupied by C (1) . The remaining servers and customer types now behave like a subsystem…”

“…for all subsets, then L = 2 and the minimum of these ratios will be reached by S\S (1) , C\C (1) . Else, if the minimum is again < 1, the subsets C, S(C) with minimal ratio of β S(C)\S (1) /α C will be S (2) , C (2) , and the servers in S (2) will become overloaded when λ = µ…”