Queue Management Project Report


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A Queueing Model is a suitable model to represent a service oriented problem where customers arrive randomly to receive some services, the service time also being a random variable.

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Queue Management Project Report

  1. 1. Quantitative Techniques of Management QUEUING THEORY Made By: Neepa Sharma Swati Singhal Akanksha Goel Aditi Gupta Batch of 2012 SSCBS, DU Submitted to: Mr. Neeraj Kumar
  2. 2. Queuing Theory BasicsCommunication DelaysBefore we proceed further, let us understand the different components of delay in a messagingsystem. The total delay experienced by messages can be classified into the following categories:Processing Delay  This is the delay between the time of receipt of a packet for transmission to the point of putting it into the transmission queue.  On the receive end, it is the delay between the time of reception of a packet in the receive queue to the point of actual processing of the message.  This delay depends on the CPU speed and CPU load in the system.Queuing Delay  This is the delay between the point of entry of a packet in the transmit queue to the actual point of transmission of the message.  This delay depends on the load on the communication link.Transmission Delay  This is the delay between the transmissions of first bit of the packet to the transmission of the last bit.  This delay depends on the speed of the communication link.Propagation Delay  This is the delay between the point of transmission of the last bit of the packet to the point of reception of last bit of the packet at the other end.  This delay depends on the physical characteristics of the communication link.Retransmission Delay  This is the delay that results when a packet is lost and has to be retransmitted.  This delay depends on the error rate on the link and the protocol used for retransmissions.
  3. 3. What we would be discussing in detail is the queuing delay and how to deal with it.Meaning of a Queuing ModelA Queuing Model is a suitable model to represent a service oriented problem where customers arriverandomly to receive some services, the service time also being a random variable.Objectives of a Queuing ModelThe objective of a Queuing Model is to find out the optimum service rate and the number of servers sothat the average cost of being in Queuing system and the cost of service are minimized.The Queuing Problem is identified by the presence of a group of customers who arrive randomly toreceive some service. The customer upon arrival may be served immediately or if willing may have towait until the server is free.Application of a Queuing ModelThe Queuing Models are basically relevant to service oriented organizations and suggest ways andmeans to improve the efficiency of the service. Thos model can be applied in the field of business(banks, booking counters), industries (servicing of machines), government (railway or post-officecounters), transportation (airport, harbor) and everyday life (elevators, restaurants, and doctor’s clinic).Relationship between SERVICE and COSTAn improvement of service level is always possible by increasing the number of employees. Apart fromincreasing the cost an immediate consequence of such a step is unutilized or idle time of the servers. Inaddition, it is unrealistic to assume that a large-scale increase in staff is possible in an organization.Queuing Methodology indicates the optimum usage of existing manpower and other resources toimprove the service. It can also indicate the cost implication if the existing service facility has to beimproved by adding more servers.
  4. 4. The relationship between queuing and service rates can be diagrammatically illustrated using the costcurves as shown in the following figure. Cost of Operation Cost of Service COST Cost of Waiting SLOW TIME QUICKAt a slow service rate, queues build up and the cost of queuing increases. An ideal service unit willminimize the operating cost of the entire system.ARRIVALThe statistical pattern of the arrival can be indicated through- I. The probability distribution of the number of arrivals in a specific period of time, or II. The probability distribution of the time between two successive arrival (known as inter arrival time)The number of arrivals is a discrete variable where as the inter arrival times are continuous randomand variable.
  5. 5. A remarkable result in this context is that if the number of arrivals follows a ‘Poisson Distribution’, thecorresponding inter arrival time follows an ‘Exponential Distribution’. This property is frequently used todelegate elegant results on queuing problems.SERVICEThe time taken by a server to complete service is known as service time. The service time is a statisticalvariable and can be studied either as the number of services completed in a given period of time or thetime taken to complete the service. The data on actual service time should be analysed to find out theprobability distribution of service time. The number of services completed is a discrete random variablewhile the service time is a continuous random variable.SERVERA server is a person or mechanism through which service is offered. The service may be offered througha single server such as a ticket counter or through several channels such as a train arriving in a stationwith several platforms.Sometimes the service is to be varied out sequentially through several phases known as multiphaseservice. In government, the papers move through a number of phases in terms of official hierarchy tillthey arrive at the appropriate level where a decision can be taken.Time Spent in The Queuing SystemThe time spent by a customer in a Queuing System is the sum of waiting time before service and theservice time. The waiting time of a customer is the time spent by a customer in the Queuing Systembefore the service starts. The probability distribution of waiting time depends upon the probabilitydistribution of inter arrival time and service time.Queue DisciplineThe queue discipline indicates the order in which members of the queue are selected for service. It ismost frequently assumed that the customers are served on first come first serve basis. This is commonlyreferred to as FIFO (first in, first out) system.Occasionally a certain group of customers receive priority over the others even if they arrive late. This iscommonly referred to as Priority Queue.The queue discipline does not always take into account the order of arrival. The server chooses one ofthe customers to offer service at random. Such a service is known as service in random order (SIRO).While allotting an item with high demand and limited supply such as a test match ticket or share of a
  6. 6. public limited company, SIRO system is the only pay of offering service when it is not possible to identifythe order of arrival.It may also be so that every customer gets a time slice. If her service is not completed within that slice oftime, he/she will re-enter the queue. This is known as RoundRobin.State of Queuing SystemThe transient state of a Queuing System is the state where the probability of the number of customersin the system depends upon time.The steady state of a Queuing System is the state where the probability of the number of customers inthe system is independent of time.Fundamental Components of a Queuing ProcessThe fundamental components of a queuing process are listed below:  The input process or the arrivals.  Service mechanism  Queue disciplineWe now give a brief description of each of the above components: 1. The input process: Customers arrive at a service station for service. They don’t come at regular intervals but arrivals into the system occur according to some chance mechanism. Often an arrival occurs at random and is independent of what has previously occurred. Customers may arrive for service individually or in groups. Single arrivals are illustrated by customers visiting a bank. On the other hand, families visiting a restaurant, is an example of bulk or group arrival. Arrivals may occur at a constant rate or may be in accordance with some probability such as poisson distribution or normal distribution, etc. Frequently the population of the units or customers requiring service may be infinite e.g. passengers waiting across the booking counters but there are situations where the population may be limited such as the number of particular equipment breaking down and requiring service from the factor’s maintenance crew.
  7. 7. Thus, in connection with the input process, the following information is usually called for and is considered relevant: a) Arrival distribution b) Inter-arrival distribution c) Mean arrival rate (or the average number of customers arriving in one unit of time) d) Mean time between intervals.2. Service Mechanism: Analogous to the input process, there are probability distribution of service times and number of customers served in an interval. Service time can either be fixed (as in the case of a vending machine) or distributed in accordance with some probability distribution. Service facilities can be any one of the following types: a) Single channel facility or one queue-one channel service station facility – This means that there is only one queue in which the customer waits till the service pint is ready to take him for servicing. A library counter is an example of this. b) One queue-several service station facilities – In the case customers wait in a single queue and the customer can join any one of the service stations is ready to take them for servicing. Booking at a service station that has several mechanics each handling one vehicle, illustrates this type of model. c) Several queues-one service station – In such a situation, there are several queues and the customer can join any one of these but the service station is only one. d) Multi-channel facility – In this model, each of the servers has a different queue. Different cash counters in an Electricity Board Office, where the customers can make payment in respect of their electricity bills, provides an example of this model. Booking counters at railway station provide another example. e) Multi-stage multi-channel facilities – In this case, customers require several types of service and different service stations are there. Each station provides a specialized service and the customer passes through each of the several stations before leaving the system. For example, machining of a certain steel item may consist of cutting, turning, knurling, drilling, grinding and packaging, etc., each of which is performed by a single server in a series.
  8. 8. In connection with the service mechanism the following information often obtained from the point of view of the queuing theory: i. Distribution of number of customers serviced ii. Distribution of time taken to service customers iii. Average number of customers being serviced in one unit of time at a service station iv. Average time taken to service a customer. 3. Queue Discipline Queue discipline may refer to many things. One of such things is the order in which the service station selects the next customer from the waiting line to be served. In this context, queue discipline may be like first in, first out or last in, first out or may be on the basis of certain other criteria. For example, it may be random when a teacher picks up the students for recitation. Sometimes the customer may be given a priority basis for service as on the basis of ladies first. Another aspect of queue discipline is whether a customer in a queue can move to a shorter queue in the multi-channel system. Queue discipline also refers to the manner in which the customers form into queue and the manner in which they behave in the queue. For example, a patient may get impatient and leave the queue.Limitations of SINGLE CHANNEL QUEUING MODELThe single channel queuing model, is the simplest model which talks of a single service station attendingarrivals from infinite population, serving customers on a first come, first serve basis.However, in reality, there are several limitations of this model in its application. On obvious limitation isthe possibility that the waiting space may in fact be limited. Another possibility is that arrival rate isstate dependent. That is, potential customers are discouraged from entering the queue if they observe along line when they arrive. Another practical limitation of the model is that the arrival process is notstationary. It is quite possible that the service station would experience peak periods and slack periodsduring which the arrival rate is higher and lower respectively than the overall average. These could occurat particular times during a day or a week or particular weeks during a year. There is not a great dealone can do to account for stationarity without complicating the mathematics enormously. Thepopulation of customers served may be finite, the queue discipline may not be first come, first serve. Ingeneral, the validity of these models depends upon stringent assumptions that are often unrealistic inpractice.
  9. 9. Also, queuing models give steady state solution, that is, the models tell us what will happen afterqueuing system has been in operation long enough to eliminate the effects of starting with an emptyqueue at the beginning of each business day. In some applications. The queuing system never reaches asteady state, so the model solutions are of little value.Limitations of Queuing TheoryThe assumptions of classical queuing theory may be too restrictive to be able to model real-world situations exactly. The complexity of production lines with product-specific characteristicscannot be handled with those models. Therefore specialized tools have been developed tosimulate, analyze, visualize and optimize time dynamic queuing line behavior.For example; the mathematical models often assume infinite numbers of customers, infinitequeue capacity, or no bounds on inter-arrival or service times, when it is quite apparent that thesebounds must exist in reality. Often, although the bounds do exist, they can be safely ignoredbecause the differences between the real-world and theory is not statistically significant, as theprobability that such boundary situations might occur is remote compared to the expected normalsituation. Furthermore, several studies show the robustness of queuing models outside theirassumptions. In other cases the theoretical solution may either prove intractable or insufficientlyinformative to be useful.APPLICATION OF QUEUING MODEL TO INVENTORY PROBLEMSQueues are common feature in inventory problem. We are confronted with queue-like situations instores for spare parts in which machines wait for components and spare parts in service station. We canalso look at the flow of materials as inventory queues in which demands wait in lines, converselymaterials also wait in queues for demands to be served. If there is a waiting line of demands, inventorystate tends to be higher than necessary. Also, if there is a negative state of inventories, then demandsform a queue and remain unfulfilled. Thus, the management is faced with the problem of choosing a combination of controllablequantities that minimize losses resulting from the delay of some units in the queue and the occasionalwaste of service capacity in idleness. An increase in the potential service capacity will reduce the
  10. 10. intensity of congestion, but at the same time, it will also increase the expense due to idle facilities inperiods of ‘NO demand’. Therefore, the ultimate goal is to achieve a balance between the cost of serviceand the cost associated with the waiting of that service. Queuing theory contributes vital informationrequired for such a decision by predicting various characteristics of the waiting line, such as averagelength of queue. Based on probability theory, it attempts to minimize the extent and duration of queuewith minimum of investment on inventory and service facilities. Further, it gives the estimated averagetime and intervals under sampling methods, and helps in decision of optimum capacity so that the costof investment in minimum keeping the amount of queue within tolerance limits.What is Waste?There are 3 types of activities, 2 of which produce waste: 1. Steps that definitely create value. 2. Steps that create no value, but are necessary given the current state of the system. 3. Steps that create no value and can be eliminated. 4.(2) & (3) naturally create wastes, of which there are 7 types: 1. Over-Production: Producing more than is needed, faster than needed or before needed. 2. Wait-time: Idle time that occurs when co-dependent events are not synchronized. 3. Transportation: Any material movement that does not directly support immediate production. 4. Processing: Redundant effort (production or communication) which adds no value to a product or service. 5. Inventory: Any supply in excess of process or demand requirements. 6. Motion: Any movement of people which does not contribute added value to the product or service. 7. Defect: Repair or rework of a product or service to fulfill customer requirements.Time-StudyTime-study analysis is common in industrial engineering. Conducting a time-study helps toreveal any waste or problems related to the service or product being used. Time-studies are usedby industrial engineers, usability analyst, and also by ethnographers to learn about how peopleuse products and how long it takes people to do something. The data gained from a time-studycan be invaluable and can help the firm improve their product, service, or overall operation.
  11. 11. How This Relates to Queuing TheoryEliminating waste reduces time that things are in process, allowing for other items to enter thequeue. Reducing waste and time-traps helps the servers in the queue complete jobs and allowsthe jobs to exit the queue, freeing resources so that others can enter.ConclusionTravel time is necessary, but some of it can be reduced or completely eliminated. Travel timeand some processing time are elements that customers would rather not pay for — it’s a burdenthat they shouldn’t have to carry in the form of higher prices or defects. We can be proactive inidentifying waste in our processes — in any process — and employing process improvement tounlock the value-add that is in our business, resulting in a better customer experience, lowercosts, and perhaps a more profitable business.Psychology of queuesThere are a few key behavioral responses or reactions to queues, or waiting. Below are thepropositions: 1. Unoccupied time feels longer than occupied time. 2. Process-waits feel longer than in-process waits. 3. Anxiety makes waits seem longer. 4. Uncertain waits seem longer than known, finite waits. 5. Unfair waits are longer than equitable waits. 6. The more valuable the service, the longer the customer is willing to wait. 7. Solo waits feel longer than group waits.
  12. 12. Call Centers as Queuing SystemsIt’s clear that a Call Center is a Queue.The flow of calls begins with K trunk lines that connect to the Call Center. There are w ≤ k workstations, or seats, at which N ≤ w agents serve incoming calls. An arriving call that finds all ktrunk lines occupied (let’s assume there are 8 trunk lines), receives a busy signal and is blockedfrom entering the system. Else, it is connected to the Call Center and occupies one of the freelines. If fewer than N agents are busy, then the call is routed directly to an agent and is served.But, if there are more calls in the system and all agents are occupied, then the call waits in thequeue until an agent serves a call and that call exits the system. Once the call is served, theresources are released — trunk line, agent, work station.Some calls are considered retrials, because they are not served and attempt to re-enter thesystem. For some calls, the customer voluntarily abandons the call and does not call back. Also,customers that are served may attempt to re-enter the system; this happens sometimes if therewas no first call resolution (for customer service centers) and/or if the customer needed to ordersomething else (for inbound order centers).Given the simple system above, the number of trunk lines acts as an upper bound (k). Thenumber of agents acts as an upper bound on how many simultaneous calls can be in servicesimultaneously. A Call Center manager dynamically changes the number of agents on and off thefloor based on expected demand and staffing availability.Call Centers and other operation centers like it are highly complex and dynamic. The modelabove is a very simple system (read: simplistic). But, it’s easy to see how Call Centers areQueues and all the tools that Queuing Theory makes available to us can be applied to CallCenters.