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Data Structures( 数据结构 ) Course 5:Queue

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Data Structures(数据结构)Course 5:Queue

queue 队列

enqueue 进队

dequeue 出队

queue front 队头

queue rear 队尾

Queuing theory排队论

- 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

- Enqueue: Enqueue inserts an element at the rear of the queue.

grape

data

Enqueue

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kiwi

grape

plum

kiwi

front

rear

front

rear

Queue

Queue

Operation

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

plum

data

Dequeue

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kiwi

plum

kiwi

grape

front

rear

front

rear

Operation

Queue

Queue

- Queue Front:Queue front examines the element at the front of the queue.

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data

Queue

front

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kiwi

grape

front

rear

front

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Operation

Queue

Queue

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

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data

Queue

rear

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grape

plum

kiwi

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front

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front

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Operation

Queue

Queue

- 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 .

plum

kiwi

grape

fig

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fig

front

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Conceptual queue

rear

front

4

front

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Physical queue

data

next

count

front

rear

Head structure

Node structure

- Queue Algorithms
- Create queue: set the metadata pointers to null and the count to 0.

No queue

count

count

front

rear

front

rear

0

?

?

?

Before

After

- Algorithm createQueue (ref queue <metadata>
- queue.fornt = null
- Queue.rear = null
- Queue.count = 0
- End createQueue

- 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.

count

count

front

rear

front

rear

0

1

data

data

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next

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newPtr

newPtr

After

Before

Insert into empty queue

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rear

newPtr

1

data

data

next

next

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count

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newPtr

2

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data

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next

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- 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
- End if
- Queue.rear = newPtr
- Queue.count = queue.count + 1
- Return true
- End enqueue

There are four ways to test if the queue is null 1.Front null

2.Rear null

3.Count 0

4.Emptyqueue

Before

After

Insert into queue with data

count

front

rear

1

data

next

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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.

count

front

rear

0

(recycled)

deleteLoc

Before

After

Delete only item in queue

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- 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
- End if
- Queue.front = queue.front->next
- Queue.count = queue.count – 1
- Recycle (deleteLoc)
- Return true
- End dequeue

Before

count

front

rear

1

data

data

next

next

plum

kiwi

(recycled)

After

deleteLoc

- 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

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

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

- 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.

- A Single-server queue can provide service to only one customer at a time.

- 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.

- 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?

Queue

Server

Queue

time

Service

time

Response

time

A queuing theory model

- 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)

- 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

- 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.

count

front

rear

2

custNum

arriveTime

next

startTime

svcTime

arriveTime

custNum

totWaitTime

maxQueueSize

totSvcTime

numCust

head

node

custStatus

simStats

Figure 5-13 queue data structures

- 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

stats

Figure 5-14 design for queue simulation

- Simulation Algorithm
- Simulator

custStatus

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

- 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

- 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

- 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

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

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

- 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

3 22 12 6 10 34 65 29 9 30 81 4 5 19 20 57 44 99

3 6 9 4 5 12 10 19 22 29 20 34 65 30 81 57 44 99

Algorithm categorize

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

- Algorithm fillQueues (ref q0to9 <metadata>,
- ref q10to19 <metadata>,
- ref q20to29 <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

- 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.

- 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.

- 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.

- 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)

- 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

End loop

The data are 5,7,12,4,0,4,6