$j.ajax({
        type: "POST",
        url: "/Services/Cart.asmx/UpdateQuantity",
        data: data,
        contentType: "application/json; charset=utf-8",
        dataType: "json",
        success: function(msg) {
            var d = msg.d;

            // etc.
        },
        error: function(xhr, msg) {
            var response = JSON.parse(xhr.responseText);
            var cart = $j('#cart');
            if (response.Message) {
                cart.before(' <div class="error">' + response.Message + '</div> ');
            }
            else {
                cart.before(' <div class="error">An unexpected error occurred.</div> ');
            }
        },
        complete: function(request, settings) {
            // Some other stuff.
        }

    }); // end ajax
    alert('Initially thought this would be called after above .ajax call complete. But it is not. Rather, it is called before success etc.')

$j.ajax({
    type: "POST",
    url: "/Services/Cart.asmx/UpdateQuantity",
    data: data,
    contentType: "application/json; charset=utf-8",
    dataType: "json",
    success: newSuccessFunction(xhr,msg),
    error: function(xhr, msg) {
        var response = JSON.parse(xhr.responseText);
        var cart = $j('#cart');
        if (response.Message) {
            cart.before(' <div class="error">' + response.Message + '</div> ');
        }
        else {
            cart.before(' <div class="error">An unexpected error occurred.</div> ');
        }
    },
    complete: function(request, settings) {
        // Some other stuff.
    }

}); // end ajax

function newSuccessFunction(xhr,msg) {
    var d = msg.d;
    //... etc.
    $.ajax({ //...
           });
    alert('Initially thought this would be called after above .ajax call complete. But it is not. Rather, it is called before success etc.')
}