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**Data Structures(数据结构) Course 5:Queue**

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Vocabulary queue 队列 enqueue 进队 dequeue 出队 queue front 队头 queue rear 队尾 Queuing theory排队论

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5.1 Queue Operations A queue is a linear list in which data can be inserted at one end, called the rear, and deleted from the other end, called the front. It is a first in-first out (FIFO) data structure. Remove (dequeue) (Enqueue) front rear A computer queue

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**Dequeue: Dequeue deletes an element at the front of the queue.**

Enqueue: Enqueue inserts an element at the rear of the queue. grape data Enqueue plum kiwi plum kiwi grape front rear front rear Queue Queue Operation Dequeue: Dequeue deletes an element at the front of the queue. plum data Dequeue plum kiwi grape kiwi grape front rear front rear Queue Operation Queue

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**Queue Rear: Queue rear examines the element at the rear of the queue.**

Queue Front: Queue front examines the element at the front of the queue. plum data Queue front plum kiwi grape plum kiwi grape front rear front rear Operation Queue Queue Queue Rear: Queue rear examines the element at the rear of the queue. grape data Queue rear plum kiwi grape plum kiwi grape front rear front rear Operation Queue Queue

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**5.2 Queue Linked List Design**

Data structure: For the linked list implementation of a queue, we use tow types of structures: a head and a node. Queue head: The queue head contains the two pointers and a count of the queue. Queue data node: The queue data node contains the user data and a link field pointing to the next node .

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**front rear front count rear data next plum kiwi grape fig front rear**

Conceptual queue front rear 4 plum kiwi grape fig front rear Physical queue front count rear data next Head structure Node structure

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Queue Algorithms Create queue: set the metadata pointers to null and the count to 0. No queue front count rear front count rear ? ? ? Before After Algorithm createQueue (ref queue <metadata> queue.fornt = null Queue.rear = null Queue.count = 0 End createQueue

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**Enqueue: Three conditions need to be considered: **

1.insert into an empty queue. 2. Insert into a queue with data. 3. Insertinto a queue when there is no memory left in the heap. front count rear front count rear 1 data next data next newPtr plum plum newPtr After Before Insert into empty queue

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**plum next data front rear count kiwi plum next data 1 front rear count**

Algorithm enqueue (ref queue<metadata> dataIn <dataType> If (queue full) 1 return false End if Allocate (newPtr) newPtr->data = dataIn newPtr->next = null pointer If (queue.count zero) // inserting into null queue 1 queue.front = newPtr Else // insert data 1 queue.rear->next = newPtr Queue.rear = newPtr Queue.count = queue.count + 1 Return true End enqueue plum next data 1 front rear count kiwi newPtr There are four ways to test if the queue is null 1.Front null 2.Rear null 3.Count 0 4.Emptyqueue Before plum next data 2 front rear count kiwi newPtr After Insert into queue with data

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**Dequeue: 1. Ensure that the queue contains data. 2**

Dequeue: 1. Ensure that the queue contains data. 2. Pass the data back through the parameter list and then set the front pointer to the next item in the queue. 3. If the queue is now empty, set the rear pointer to null. plum next data 1 front rear count front count rear (recycled) deleteLoc Before After Delete only item in queue

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**plum next data front rear count kiwi front count rear data data next**

2 front rear count kiwi Algorithm dequeue (ref queue <metadata> ref item <dataType>) If (queue.count is 0) 1 return false End if Item = queue.front->data deleteLoc = queue.front If (queue.count 1) // Delete only item in queue 1 queue.rear = null pointer Queue.front = queue.front->next Queue.count = queue.count – 1 Recycle (deleteLoc) Return true End dequeue Before front count rear 1 data next data next plum kiwi (recycled) After deleteLoc

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Retrieving Queue Data: the logic of retrieving data is the same to that of dequeue except that the data are not deleted from the queue. Algorithm queueFront ( val queue <metadata>, ref dataOut <dataType>) If (queue.count is 0) 1 return false End if dataOut = queue.front->data Return true End queueFront

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**Empty Queue: it returns true if the queue is empty and false if the queue contains data.**

Algorithm emptyQueue ( val queue <metadata>) Return (queue.count equal 0) End emptyQueue Full Queue: By allocating a node and then releasing the memory we can determine whether there is room for at least one more node. Algorithm fullQueue ( val queue <metadata>) Allocate (tempPtr) If (allocate successful) 1 recycle (tempPtr) 2 return false Else 1 return true End if End fullQueue

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Queue Count: it returns the number of elements currently in the queue by returning the count found in the queue head node. Algorithm Queuecount ( val queue <metadata>) Return (queue.count) End queueCount Destroy Queue: it deletes all data in the queue and recycles their memory. Algorithm destroyQueue ( ref queue <metadata>) pWalker = queue.front Loop (pWalker not null) 1 deletePtr = pWalker 2 pWalker = pWalker.next 3 recycle (deletePtr) End loop Queue.front = null Queue.rear = null Queue.count = 0 return End destroyQueue

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5.3 Queuing Theory Queuing theory is a field of applied mathematics that is used to predict the performance of queues. A Single-server queue can provide service to only one customer at a time. Example: the hot-food vendor. A Multi-server queue can provide service to many customers at a time. Example: a bank in which there is one line with many bank tellers providing service.

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**Two elements to all queues**

A customer is any person or thing needing service. Such as jobs in computer, packages being sent… The service is any activity needed to accomplish the required result. Two factors affect the queue The arriving rate(比率) is the rate at which customers arrive in the queue for service. Depending on the service being provided, the arrival rate may be random or regular. Service time is the average time required to complete the processing of a customer request. The arriving rate and service time are the factors that most affect the performance of queues.

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**The faster customers arrive and the higher the service time, the longer the queue will be.**

The ideal is arrival rate matches service time The importance of queuing theory: it can predict the queue patterns including queue time(that is, the average length of time customers wait in the queue), the average size of the queue, and the maximum queue size. So, we can build a model of queue and used the model to study proposed changes to the system. For example, In the banking queue, if we were able to add automation improvements that would reduce the average service by 15%,how many fewer tellers would we need?

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**Queue time Service time Response time Queue Server**

A queuing theory model

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5.4 Queue Applications Two queue implementations: Queue simulation and categorizing data Queue simulation: a modeling activity used to generate statistics about the performance of queues. An example: a saltwater taffy store on a beach boardwalk. The store has one window and a clerk can service only one customer at a time. The store also ships boxes of taffy anywhere in the country.The time to serve customers varies between 1 and 10 minutes.(8hs per day, 7 days a week)

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Events: completed process new customer module: determine the arrival of a new customer. The owner found that, on average , a customer arrives every 4 minutes. An arrival rate is simulated by using a random number generator that returns a values between 1 and 4. If = 4, customer arrived; 1,2,3 customer not arrived. server free module: determine whether the clerk is busy or idle. If the clerk is idle, then the next waiting customer in line can be served. If the clerk is busy, then the waiting customers remain in the queue. Completed processing: determine whether it has completed processing for the current customer. Then processing time for the current customer is determined by a random number generator when the processing is started. When customers has been completely served, we gather statistics about sale and set server to an idle state

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Data structures: Four data structure are required for the queue simulation Queue head: It contains two node pointers – front and rear – and a count of the number of elements currently in the queue. Queue node: It contains the customer data and a next node pointer. The customer data consist of a sequential customer number and the arrival time. Current Customer status: We use customer’s number, arrival time, the start time and the processing time to describe customer status.(random generator to calculate) Simulation statistics: It stores the total number of customers processed in the simulation, the total and average service time, the total and average wait time, and the maximum number of customers in the queue at one time.

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**front rear count head next node custNum arriveTime startTime svcTime**

2 front rear count head next custNum arriveTime node custNum arriveTime startTime svcTime custStatus numCust totSvcTime totWaitTime maxQueueSize simStats Figure 5-13 queue data structures

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Output: the statistics gathered during the simulation and the average queue wait time and average queue service time, the basic statistics for each customer: arrival time, start time, wait time, service time etc. Simulator Create queue New customer Server free Service complete Print stats Figure 5-14 design for queue simulation

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**Simulation Algorithm Simulator custStatus Algorithm taffySimulation**

custNum <integer> arriveTime <integer> startTime <integer> svcTime <integer> end custstatus simStats numCust <integer> totSvcTime <integer> totWaitTime <integer> maxQueueSize <integer> end simstats Algorithm taffySimulation Data Structures data number <integer> arrivalTime <integer> end data head front <node pointer> count <integer> rear <node pointer> end head node custData <data> next <node pointer> end node

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**Algorithm 5-9 queue simulation: driver**

Statements CreateQueue (queue) Clock = 1 endTime = 8*60 custNum = 0 Loop (clock <=endTime or moreCusts) 1 newCustomer (queue, clock, custNum) 2 serverFree (queue, clock, custStatus, moreCusts) 3 svcComplete (queue, clock, custStatus, runStats, moreCusts) 4 if ( not emptyQueue (queue)) 1 moreCusts = true 5 end if 6 clock = clock + 1 End loop printStats (runStats) return end taffySimulation Algorithm 5-9 queue simulation: driver

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**New customer Algorithm newCustomer (ref queue < metadata >,**

val clock <integer>, ref custNum <integer>) Arrival = (random number modulo 4) + 1 If (arrival equal 4) // new customer has arrived 1 custNum = custNum + 1 2 custData.number = custNum 3 custData.arriveTime = clock 4 enqueue (queue, custData) End if Return End newCustomer

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**Server free Algorithm serverFree ( ref queue <metadata>,**

val clock <integer> , ref status <custStastus>, ref moreCusts <Boolean> ) If (clock > status.startTime + status.svcTime – 1) // server is idle 1 if (not emptyQueue (queue)) 1 dequeue (queue,custData) 2 status.custNum = custData.number 3 status.arriveTime = custData.arriverTime 4 status.startTime = clock 5 status.svcTime = random service time 6 moreCusts = true 2 end if End if Return End serverFreestatus

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**Service complete Algorithm svcComplete (ref queue <metadata>,**

val clock <integer>, ref status <custStatus>, ref stats <simStats>, ref moreCusts <Boolean>) If (clock equal status.startTime + status.svcTime – 1) //current call complete 1 waitTime = status.startTime – status.arriveTime 2 stats.numCust = stats.numCust + 1 3 stats.totSvcTime = stats.totSvcTime + status.svcTime 4 stas.totWaitTime = stats.totWaitTime + waitTime 5 queueSize = queueCount (queue) 6 if (stas.maxQueueSize < queueSize) 1 stats.maxQueueSize = queueSize 7 end if 8 print ( status.custNum status.arriveTime status.startTime status.svcTime waitTime queueCount(queue)) 9 moreCusts = false Return End svcComplete

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**Print stats Algorithm printStats ( stats <simStats> )**

Print (Simulation Statistics: ) Print (Total customers: stats.numCust) Print (Total service time: stats.totSvcTime) avrgSvcTime = stats.totSvcTime / stats.numCust Print (Average service time: arvgSvcTime) avrgWaitTime = stats.totWaitTime / stats.numCust Print (Average wait time: avrgWaitTime) Print (,Maximum queue size: stats.maxQueueSize) return End printstats

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**For example, given the following list of numbers**

Categorizing Data: It is often necessary to rearrange data without destroying their basic sequence. For example, given the following list of numbers then categorize them into four different groups: Group1: less than 10 Group2: between 10 and 19 Group3: between 20 and 29 Group4: 30 and greater

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Algorithm categorize CreateQueue (q0to9) createQueue (q10to19) createQueue (q20to29) createQueue (qOver29) fillQueues (q0t09, q10to19,q20to29, qOver29) printQueues (q0to9, q10to19, q20to29,qOver29) Return End categorize

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**Algorithm fillQueues (ref q0to9 <metadata>,**

ref qOver29 <metadata>) Loop (not EOF) 1 read (number) 2 if (number < 10) 1 enqueue (q0to9, number) 3 elseif (number<20) 1 enqueue (q10to19, number) 4 elseif (number<30) 1 enqueue (q20to29, number) 5 else 1 enqueue (qOver29, number) 6 end if End loop Return End fillQueue

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5.8 Summary A queue is a linear list in which data can only be inserted at one end, called the rear, and deleted from the other end, called the front. A queue is a first in-first out (FIFO) structure. There are four basic queue operations: enqueue, dequeue, queue front, and queue rear. The enqueue operation inserts an element at the rear of the queue. The dequeue operation deletes the element at the front of the queue. The queue front operation examines the element at the front of the queue without deleting it. The queue rear operation examines the element at the rear of the queue without deleting it.

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**To implement the queue using a linked list, we use two types of structures: a head and a node.**

Queuing theory is a field of applied mathematics that is used to predict the performance of queues. Queue applications can be divided into single servers and multi-servers. A single-server queue application provides service to only one customer at a time. A multi-server queue application provides service to only several customers at a time. The two features that most affect the performance of queues are the arrival rate and the service time. The rate at which the customers arrive in the queue for service is known as the arrival rate. Service time is the average time required to complete the processing of a customer request.

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**The queue time is the average length of time customers wait in the queue.**

The response time is a measure of average time from the point at which customers enter the queue until the moment they leave the server. It is queue time plus service time. One application of queues is queue simulation, which is a modeling activity used to generate statistics about the performance of a queue. Another application of queues is categorization. Queues are used to categorize data into different groups without losing the original ordering of the data. Queues can be implemented suing linked lists or arrays.

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Exercise Imagine you have a stack of integers,S ,and a queue of integers,Q. Draw a picture of S and Q after the following operation: PushStack(S,3) PushStack(S,12) Enqueue(Q,5) Enqueue(Q,8) PopStack(S,x) pushStack(S,2) Enqueue(Q,x) Dequeue(Q,y) PushStack(S,x) PushStack(S,y)

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Exercise What would be the contents of queue Q1 and Q2 after the following code is executed and the following data are entered? Q1=createQueue Q2=createQueue Loop (not end of file) read number enqueue(Q1,number) enqueue(Q2,number) loop (Not empty Q1) dequeue(Q1,x) enqueue(Q2,x) End loop The data are 5,7,12,4,0,4,6

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