  The model is a service system, consisting of several large server pools. A
server processing speed and buffer size (which may be finite or infinite)
depend on the pool. The input flow of customers is split equally among a fixed
number of routers, which must assign customers to the servers immediately upon
arrival. We consider an asymptotic regime in which the customer total arrival
rate and pool sizes scale to infinity simultaneously, in proportion to a
scaling parameter $n$, while the number of routers remains fixed.
  We define and study a multi-router generalization of the pull-based customer
assignment (routing) algorithm PULL, introduced in [11] for the single-router
model. Under PULL algorithm, when a server becomes idle it send a
"pull-message" to a randomly uniformly selected router; each router operates
independently -- it assigns an arriving customer to a server according to a
randomly uniformly chosen available (at this router) pull-message, if there is
any, or to a randomly uniformly selected server in the entire system,
otherwise.
  Under Markov assumptions (Poisson arrival process and independent
exponentially distributed service requirements), and under sub-critical system
load, we prove asymptotic optimality of PULL: as $n\to\infty$, the steady-state
probability of an arriving customer experiencing blocking or waiting, vanishes.
Furthermore, PULL has an extremely low router-server message exchange rate of
one message per customer. These results generalize some of the single-router
results in [11].
