Review the literature on MEWMA. Some of the published articles explain this issue. It usually is some form of Markov Chain or it is done with simulation.
If you are an R user, then install my R package "spc" and call the function mewma.arl() to calculate the ARL. I also want to to promote my paper Knoth (2017), ARL Numerics for MEWMA Charts, Journal of Quality Technology 49(1), 78-89. Eventually, the methods in use are (i) Monte-Carlo simulation (utilized by the inventors of MEWMA), (ii) Markov chain approximation (Runger/Prabhu 1996 and more; free software is available), (iii) numerical solution of integral equation (Rigdon 1995; for the in-control ARL is free software available; Minitab and Statistica offer it as well).