Estimating mean sojourn time in the processor sharing M/G/1 queue with inaccurate job size information

We seek to approximate the mean sojourn time in the processor sharing M/G/1 queue with inaccurate job size information. Suppose we are given the arrival rate λ and random service time Ŝ = SX where X ∼ LN(0, σ) represents the inaccuracy. Denote the mean sojourn time in an M/G/1 queue with processor sharing with service time Ŝ by E(T̂) and with service time S by E(T). Finally, E(T̂) denotes the mean sojourn time of an M/G/1 queue with resampling service policy and service time distribution according to S. It can be shown that for exponential service time S, E(T) < E(T̂) < E(T̂) holds for any σ > 0.