  In this paper, we discuss an interesting but challenging bilateral
stochastically matching problem: A more general matched queue with matching
batch pair (m, n) and two types (i.e., types A and B) of impatient customers,
where the arrivals of A- and B-customers are both Poisson processes, m
A-customers and n B-customers are matched as a group which leaves the system
immediately, and the customers' impatient behavior is to guarantee the
stability of the system. We show that this matched queue can be expressed as a
novel bidirectional level-dependent quasi-birth-and-death (QBD) process. Based
on this, we provide a detailed analysis for this matched queue, including the
system stability, the average stationary queue lengthes, and the average
sojourn time of any A-customer or B-customer. We believe that the methodology
and results developed in this paper can be applicable to dealing with more
general matched queueing systems, which are widely encountered in various
practical areas, for example, sharing economy, ridesharing platform, bilateral
market, organ transplantation, taxi services, assembly systems, and so on.
