Admission control is a procedure to guarantee a given level of Quality of Service (QoS) by accepting or rejecting arrival connection requests. There are many studies on backlog or loss rate evaluation formulas for admission control at a single node. However, there are few studies on end-to-end evaluation formulas suitable for admission control. In a previous paper, the authors proposed a new stochastic network calculus for many flows using an approach taken from large deviations techniques and obtained asymptotic end-to-end evaluation formulas for output burstiness and backlog. In this paper, we apply this stochastic network calculus to a heterogeneous tandem network with many forwarding flows and cross traffic flows constrained by leaky buckets, and obtain a simple evaluation formula for the end-to-end backlog. In this formula, the end-to-end backlog can be evaluated by the traffic load at the bottle neck node. This result leads us to a natural extension of the evaluation formula for a single node.