CS2006 - Data Structures I

Chapter 7Queues II

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Topics

Queue Application Simulation

Comparison of List, Stack and Queue

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Simulation

Simulate the behavior of a system by constructing a mathematical or a physical model capturing the relevant information about the system

Real-word systems are called “Queuing Systems” Theory used in simulation is called “Queuing

Theory” Examples:

Computer tasks Supermarket

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Simulation

Goal of simulation: Generate statistics summarizing or predicting the

performance of systems, that could be used for improvement

Example: Bank system Server: Teller Objects being served: Customers What should be observed: Average waiting time Events: Arrivals & departures of customers Service time: Time needed for each customer

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Application: Bank SimulationMs. Simpson, President of First City Bank of Springfield, has heard her customers complain about how long they have to wait for service. Because she fears that they may move their account to another bank, she is considering whether to hire a second teller.

Ms. Simpson needs an approximation of the average time that customer has to wait for service from the only teller available.

How can she get this information?

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Application: Bank Simulation

Questions that should be answered: How many teller does the bank employ? How often customers arrive? What is the average time does a customer

have to wait before receiving service? How many employers should be hired to

improve the performance of the bank?

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Application: Bank Simulation Events:

Customer arrivals External events

Customer departures when completing transaction Internal events

Statistics needed: Average time a customer waits for service

= Total waiting time for all customers / Number of customers

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Application: Bank Simulation Assumptions:

The list of all arrival events, is already available for use in a file in the form of pairs

(Arrival time, Transaction time) The events are given in

Ascending order By arrival time

Departure events are not included in the file since it could be calculated

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Application: Bank Simulation

Event list: Contains the unprocessed arrival & departure

events Departure time:

Departure time = arrival time + transaction time

Waiting time: Time elapsed between arrival & start of transaction

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Application: Bank Simulation

Required operations: Add / remove customers

Add /remove events

Required data structures: A queue representing customers in line:

will contain arrival time & duration of transaction

A list representing the event list

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Application: Bank Simulation

The new customer always enters the queue, even if the queue is empty

When a customer is ready for service, the following operations take place: Remove the customer from the queue Delete the arrival event for the new customer from the

event list Insert the departure event into the event list

The place that an event is inserted in the event list depends on the relative time of that event

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Application: Bank Simulation Example:

If the file contains the following:Arrival time Transaction duration

20 522 423 230 3

The result of simulation will be as follows:Time Event

20 Cust1 enters bank & begins transaction22 Cust2 enters bank & stands at line end23 Cust3 enters bank & stands at line end25 Cust1 departs & Cust2 begins transaction29 Cust2 departs & Cust3 begins transaction30 Cust4 enters bank & stands at line end31 Cust3 departs & Cust4 begins transaction34 Cust4 departs

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Application: Bank Simulation Arrival Events

Indicating Arrival at the bank of a custmer One of the two thing happens:

If the teller is idle, the customer enters the line and begins the service

If the teller is busy, the customer enters the waiting line Departure Events

Indicating the departure of a customer who finish server

When there is another on the line, the new customer begins service

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Application: Bank Simulation

Pseudocode (First Draft) for simulation Determine the times at which the events occur, and process the events Increments the time by increments of 1

//InitializecurrentTime = 0Initialize the line to “no customers”while (currentTime <= time of final event){ if (an arrival event occurs at time currentTime)

process the arrival eventif (a departure event occurs at time currentTime)

process the departure event// when both arrival & departure occur at the same time,// arbitrarily process the arrival event first++currentTime} // end while

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Application: Bank Simulation Pseudocode (Second Draft)

Considers only times of relevant events (arrival & departure)

//Initialize the line to “no customers”while (event remain to be processed){ currentTime = time of next event

if (event is an arrival event )process the arrival event

elseprocess the departure event

// when both arrival & departure events occur at the same time,

// arbitrarily process the arrival event first} // end while

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Application: Bank Simulation

Example: Observations

Each arrival event generates exactly one departure event We cannot generate a departure event for a given arrival

event independent of other events. If we read the whole file of events, we will need to the

calculations that the simulation performs => It is better to store one arrival and one departure event at a

time and let the simulation perform the calculation step-by-step The event list will contain at most one event of each kind

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Application: Bank Simulation

Example: Observations

For arrival events Keep the earliest unprocessed arrival event When the event is processed, replace it with the next unprocessed

event For departure events

The next departure event depends on the customer currently served Evaluate time o next departure from

Time service begins Length of transaction

Departure time = arrival time + transaction time As soon as a customer begins service, we place the corresponding

departure event corresponding to this customer in the event list

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Application: Bank Simulation

Example: Event list structure

A Arrival time Transaction time

D Departure time

Arrival event

Departure event

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Application: Bank Simulation

Possible event list configurations Initially:

One arrival event A read from the input file Event list: A

Generally: Two events (Arrival A & Departure D)

Event list: A D (next event is arrival)or D A (next event is departure)

Special cases: If Arrival event is first and, after it is processed, the file is

empty (eof): Event list: D (input has been exhausted)

If Departure event is first and teller line is empty: Event list: A (departure leaves teller line empty)

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Application: Bank Simulation

Events are inserted in the event list the beginning or at the end depending on their arrival times

Algorithm for arrival events// Update the event listDelete the arrival event for customer C from the event listif ( new customer C begins transaction immediately)

Insert a departure event for customer C into the event list (time of event = current time + transaction length)If (not at the end of the input file)

Read a new arrival event & add it to the event list (time of event = time specified in file)

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Application: Bank Simulation Algorithm for departure events

// Update the lineDelete customer at front of queueIf (the queue is not empty)

Current front customer begins transaction// Update the event listDelete the departure event from the event listIf (the queue is not empty)

Insert into the event list the departure event for the customer now at the front of the queue (time of event = current time + transaction length)

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Application: Bank Simulation Final simulation algorithm

+simulate ( )// perform the simulationCreate an empty queue bankQueue to represent the bank lineCreate an empty event list eventListGet the first arrival event from the input file & place it in eventListwhile ( eventList is not empty){ newEvent = first event in eventList

if (newEvent is arrival event)processArrival(newEvent, arrivalFile, eventList,

bankQueue)else

processDeparture(newEvent, eventList, bankQueue)} // end while

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Application: Bank Simulation Final simulation algorithm

+processArrival( in arrivalEvent: Event, in arrivalFile: File, inout anEventList: EventList, inout bankQueue: Queue)

// Processes an arrival eventatFront = bankQueue.isEmpty() // present queue status// Update bankQueue by inserting the customer, as described in arrivalEvent,// into the queuebankQueue.enqueue ( arrivalEvent)// Update the event listDelete arrivalEvenet from anEventListIf (atFront){ // The line was empty, new customer at the front of

// the line begins transaction immediatelyInsert into anEventList a departure event corresponding to the new customer with currentime = currentTime + transaction length

} // end ifIf (not at end of input file){ Get the next arrival event from arrivalFile

Add the event – with time as specified in input file – to anEventList} // end if

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Application: Bank Simulation Final simulation algorithm

+processDeparture( in departureEvent: Event, inout anEventList: EventList, inout bankQueue: Queue)

// Processes an departure event// Update the line by deleting the front customerbankQueue.dequeue ( )

// Update the event listDelete departureEvent from anEventList

If (! bamkQueue.isEmpty){ // Customer at front of line begins transaction

Insert into the event list the departure event corresponding tothe customer now at the front of the line & hascurrentTime = currentTime + transaction length

} // end if

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Application: Bank Simulation ADT Event List Operations

+createEventList() // Create an empty event list+destroyEventList() // Destroys an event list+isEmpty(): boolean {query}

// Determines whether an event list is empty

+insert(in anEvent: Event) // Inserts anEvent into an event list so that events are ordered by time. If an arrival event & departure event have the same time, the arrival event precedes the departure event

+delete() // Deletes the first event from an event list+retrieve( out anEvent: Event)

// Sets anEvent to the first event in an event list

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Summary of Position-Oriented ADTs Lists

Operations are defined in terms of position of data items No restrictions on the position Operations:

create: Creates an empty ADT of the List type

isEmpty: Determines whether an item exists in the ADT

insert: Inserts a new item in any given position

remove: Deletes an item from a given position

retrieve: Retrieves the item in the specified position

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Summary of Position-Oriented ADTs

Stacks Operations are defined in terms of position of data items Position is restricted to the Top of the stack Operations:

create: Creates an empty ADT of the Stack type

isEmpty: Determines whether an item exists in the ADT

push: Inserts a new item into the Top position

pop: Deletes an item from the Top position

getTop: Retrieves the item from the Top position

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Summary of Position-Oriented ADTs

Queues Operations are defined in terms of position of data items Position is restricted to the Front & Back of the queue Operations:

Create: Creates an empty ADT of the Queue type

isEmpty: Determines whether an item exists in the ADT

enqueue: Inserts a new item in the Back position

dequeue: Deletes an item from the Front position

getFront: Retrieves the item from the Front position