The overlapping time distribution in the M/M/1 queue

In this paper, we analyze the distribution of the total overlapping time spent with other customers in the normal upper M Subscript lamda Baseline divided by normal upper M Subscript mu Baseline divided by 1 M λ / M μ / 1 M_λ/M_μ/1 queue with First-Come First-Served service discipline. We show that the Laplace–Stieltjes transform of the overlapping time reduces to an incomplete gamma function representation. We also calculate the transform of the joint distribution of the overlapping time and the number of overlaps. In addition, we prove a heavy-traffic limit for the total overlapping time with scaling 1 divided by left parenthesis 1 minus rho right parenthesis squared 1 / ( 1 − ρ ) 2 1/(1-ρ)² to a upper Weibull left parenthesis 1 divided by 2 comma 1 divided by mu right parenthesis Weibull ⁡ ( 1 / 2 , 1 / μ ) Weibull(1/2,1/μ) random variable. Finally, for the normal upper M Subscript lamda Baseline divided by normal upper G divided by 1 M λ / G / 1 Mλ/G/1 case, we derive the first two moments of the overlapping time.