the array into a closed circle and letting the head and tail drift around endlessly in that circle makes it unnecessary to ever move items stored in the array. If n is the size of the array, then computing indices modulo n will turn the array into a circle. This is still the conceptually simplest way to construct a queue in a high-level language, but it does admittedly slow things down a little, because the array indices must be compared to zero and the array size, which is comparable to the time taken to check whether an array index is out of bounds, which some languages do, but this will certainly be the method of choice for a quick and dirty implementation, or for any high-level language that does not have pointer syntax. The array size must be declared ahead of time, but some implementations simply double the declared array size when overflow occurs. Most modern languages with objects or pointers can implement or come with libraries for dynamic lists. Such data structures may have not specified a fixed capacity limit besides memory constraints. Queue overflow results from trying to add an element onto a full queue and queue underflow happens when trying to remove an element from an empty queue. A bounded queue is a queue limited to a fixed number of items. There are several efficient implementations of FIFO queues. An efficient implementation is one that can perform the operations—en-queuing and de-queuing—in O(1) time. Linked list A doubly linked list has O(1) insertion and deletion at both ends, so it is a natural choice for queues. A regular singly linked list only has efficient insertion and deletion at one end. However, a small modification—keeping a pointer to the last node in addition to the first one—will enable it to implement an efficient queue. A deque implemented using a modified dynamic array Queues and programming languages Queues may be implemented as a separate data type, or maybe considered a special case of a double-ended queue (deque) and not implemented separately. For example, Perl and Ruby allow pushing and popping an array from both ends, so one can use push and shift functions to enqueue and dequeue a list (or, in reverse, one can use unshift and pop), although in some cases these operations are not efficient. C++'s Standard Template Library provides a "queue" templated class which is restricted to