It sounds so simple, "the customer comes first", but enforcing this in a corporation can be surprisingly tough. Every business unit or manager will have justified reasons why you should make an exception and not do the right thing for a customer but you always have to fight off these requests. If you stay true to the customer, you will come out right in the end.
These requests can be simple things, "Put my part first", regardless of what the history has been. They can be dramatic, "The CEO wants it first!". Each of these would have a natural acquiescence but you have to remember the customer pays the bills. You don't want to get fired all the time but you have to keep the customer in mind.
Place the customer's needs first, satisfy their needs, and you will do well. Fulfill their desires and you will excel. The customer has problems to solve. They have a scarcity of resources, a shortage of time, and a demand from their customers. If you can keep this in mind as you design your fulfillment systems success will follow. Provide the services you can, save your customer's precious time, and look for ways to make their job easier for their end customers and success awaits!