What Is Carbon Monoxide Poisoning

Carbon monoxide (or CO) is an odorless, tasteless, invisible gas that is sometimes called the  silent killer  because it poisons and kills many people each year, without them ever being aware of the danger. Heres a look at how carbon monoxide can kill you, the risk factors, and how to detect carbon monoxide and prevent injury or death. Why You Are at Risk From Carbon Monoxide Poisoning Carbon monoxide cannot be heard, smelled, or tasted, but its produced by virtually every item in your home or garage that burns fuel. Particularly dangerous are automobile fumes in an enclosed garage or a closed car. By the time youre aware that something is wrong, theres a good chance you wont be able to function well enough to open a window or leave the building or car. How Carbon Monoxide Kills You When you breathe in carbon monoxide, it enters your lungs and binds to the hemoglobin in your red blood cells. The problem is that hemoglobin binds to carbon monoxide over oxygen, so as the level of carbon monoxide increases, the amount of oxygen your blood carries to your cells decreases. This leads to oxygen starvation or hypoxia. At low concentrations, the symptoms of carbon monoxide poisoning resemble the flu: including headaches, nausea, and fatigue. Continued exposure or higher concentrations can lead to confusion, dizziness, weakness, drowsiness, severe headache, and fainting. If the brain doesnt get enough oxygen, carbon monoxide exposure can lead to unconsciousness, coma, permanent brain damage, and death. The effects can become deadly within minutes, but long-term low-level exposure is not uncommon and leads to organ damage, disease, and a slower death. Infants, children, and pets are more susceptible to the effects of carbon monoxide than adults, so they are at greater risk for poisoning and death. Long-term exposure can lead to neurological and circulatory system damage, even when the levels arent high enough to produce a significant effect in adults. Exposure to Carbon Monoxide Carbon monoxide naturally occurs in air, however dangerous levels are produced by any type of incomplete combustion. Examples are common in the home and workplace: Incomplete burning of any fuel, such as propane, gasoline, kerosene, natural gasAutomobile exhaust fumesTobacco smokeBlocked or faulty chimneysBurning any fuel in an enclosed spaceImproperly functioning gas appliancesWood-burning stoves How to Prevent Carbon Monoxide Poisoning The best protection against carbon monoxide poisoning is a carbon monoxide alarm, which alerts you whenever carbon monoxide become elevated. There are detectors designed to sound before CO levels become dangerous and there are detectors that tell you how much carbon monoxide is present. The detector and alarms should be placed anywhere there is a risk of carbon monoxide build-up, including rooms with gas appliances, fireplaces, and garages. You can reduce the risk of carbon monoxide building to critical levels by cracking a window in a room with a gas appliance or fire, so fresh air can circulate.

Financial Performance Of Mangalore Refinery And Petrochemicals Finance Essay - Free Essay Example

Sample details Pages: 8 Words: 2494 Downloads: 5 Date added: 2017/06/26 Category Finance Essay Type Narrative essay Did you like this example? INTRODUCTION Mangalore Refinery and Petrochemicals Limited (MRPL) and Reliance Petroleum Limited (RPL) were the first two refineries established by the private sector in India. In March 1992, MRPL brought out a public issue of shares, and in September 1993, RPL did the same. Both these refineries were established at a time when the administered pricing mechanism (APM)[1]was in force. Don’t waste time! Our writers will create an original "Financial Performance Of Mangalore Refinery And Petrochemicals Finance Essay" essay for you Create order APM involved full government control over the oil and natural gas sector, where only four major government owned oil companies (IOC, HPCL, BPCL and IBP) had the right to directly market petroleum products (Refer Exhibit I). The government refineries were not able to meet the increasing demand for petroleum products. Hence, opening up of the oil and natural gas sector to private companies and dismantling APM were considered as methods for reducing the demand-supply gap of petroleum products. When the Government of India (GOI) approved private sector participation in the oil refining and petroleum industry, a new investment opportunity was made available to Indian investors. Those who invested in MRPL and RPL were optimistic about the returns on shares of both these companies since reputed leading business houses such as the Aditya Birla Group (ABG)[2]and the Reliance Group[3]promoted these refinery projects. Due to the dearth of oil company stocks promoted by the private sector, the shares of both these companies were lapped up by public investors and financial institutions. Both the public issues were heavily oversubscribed. However, few investment analysts expressed their reservations about investing in stand-alone refineries like MRPL and RPL since they felt that the financial performance of companies in the refining industry was completely dependant on the crude oil prices. In March 2002 Reliance group approved the merger of RPL with Reliance Industries Ltd. (RIL)[4]. The appointed date of merger was April 2001. Once again in April 2006 Reliance Group came out with an initial public offer (IPO) for RPL. In this case an analysis of two oil refining companies viz. MRPL and RPL (2005)[Merged] for understanding risk and return involved in investment. MANGALORE REFINERY AND PETROCHEMICALS LTD. (MRPL): Mangalore Refinery and Petrochemicals Limited (MRPL) was incorporated on 7th March 1988. Company was started as joint venture of Hindustan Petroleum Corporation (HPCL)[5]and Indian Rayon Industries Limited (IRIL) Associates (AV Birla Group). MRPL has the distinction of being the only refinery in India with two CCRs which produces unleaded petrol of high octane. In the year 1993 MRPL made a public issue consisting 4,31,60,000 shares of which 16 percent was Secured Redeemable Partly Convertible Debentures (PCDs) priced at Rs 135 which fetched 582.66 crores. It also issued 2,80,00,000 shares of which 17.5 percent was Secured Redeemable Non Convertible Debentures of Rs 200 each (with Detachable Equity Warrants) fetching Rs 560 crores. In the same year MRPL also tied up with internationally reputed technology suppliers for process technologies. In March 2003 ONGC acquired the share holding of 37.39 percent which was held by AV Birla Group. Further ONGC added the capital of Rs 600 crores and made MRPL as a majority held subsidiary of ONGC. As on 31st March 2010 HPCL holds 16.96 percent (297153518 shares) and ONGC 71.63 percent (1255354097 shares) shares of MRPL. In September 1995 MRPL started a 45 MW cogeneration plant and also three million tones refinery by the end of 1995- 96 financial year. In 1999 MRPL signed deal with Chevron- Texaco for crude sourcing. The cost effective process of debottlenecking of some units resulted in enhancing the refining capacity to 12 million tones. The refining capacity was increased to 9 MMTPA in 2001. In 2006 MRPL mad an alliance with Abu Dhabi firm and also an agreement with Mauritius Company. The refinery up gradation and expansion project was undertaken at a cost of Rs 7943 crores in 2006-07. ONGC Mangalore Petrochemicals Ltd. (OMPL), a joint venture of ONGC and MRPL was incorporated for Aromatics Project worth of Rs. 4852 crore. In 2007 MRPL entered contract with State Trading Corporation (STC), Mauritius for supplying petroleum products and 4 year product supply agreement with Shell India Marketing. MRPL and Shell Aviation in 2008 entered an agreement for a joint venture to market and supply aviation fuel. FINANCIAL PERFORMANCE Table 1 shows the financial performance of MRPL for last five years (i.e. from 2005- 06 to 2009- 10). Total income has increased consistently in first four years from Rs 2,50,443.20 million in 2005- 06 to Rs 3,84,303.90 million in 2008- 09. But it declined by 15.25 percent to Rs 3,25,670.80 million in 2009- 10. Profit before depreciation and tax (PBDT) also has followed the same trend as in the case of total income, which increased in first four years before it declined by 5.14 percent in 2009- 10. Net profit increased in first three years and declined in last two years. Net profit declined by 6.26 percent in 2008- 09 in comparison with 2007- 08 and by 6.72 percent in 2009- 10 in comparison with 2008- 09. Operating profit margin (OPM) and net profit margin (NPM) have increased over the years and company has sustained it in the last three years at a higher level compared to first two years. MRPL has also sustained the level of earnings per share (EPS) and cash earnings per share (CEPS ) albeit declining marginally since 2007- 08. Depreciation has increased consistently in all five years. Even reserves also has followed the same trend. It has increased by 29.2 percent in 2009- 10 in comparison with 2008- 09. RECENT QUARTERLY PERFORMANCE Recent quarterly performance is shown in EXHIBIT V. In case of MRPL, for 4th quarter of 2009-10 financial year reported net profit of 2,530.7 million, 58.3 percent decline from corresponding quarter of the previous year (6,076.2 million). Total income increased by 35 percent as against 50 percent increase in expenditure. Operating profit margin (OPM) was 5.75 percent as against 15.42 percent whereas net profit margin (NPM) was 2.85 percent as against 9.26 percent in the corresponding quarter of the previous year. Profit for 3rd quarter of 2009-10 financial year was at Rs. 2595.4 million compared to loss of Rs. 2854.1 million in the corresponding quarter of the previous year. Operating margin was 4.8 percent and operating profits of Rs 4486.2 million as against negative 4.7 percent operating margin and loss of 3511.7 million in the corresponding quarter of the previous year. Other income increased by 43 percent (Rs. 696.6 million) and interest cost fell by 20 percent. Company also registered a marginal rise in depreciation (3 percent). These factors led to profit before tax (PBT) of Rs 2838.3 million as against loss of 4348.6 million in the corresponding quarter of the previous year. The final outcome for the company was profit of Rs. 2595.4 million as against the loss of Rs. 2854.1 million in the corresponding quarter of the previous year. RELIANCE PETROLEUM LTD. (RPL) [2005] (MERGED): Reliance Petroleum Ltd. (RPL) was incorporated in October 2005 as a subsidiary of Reliance Industries Ltd. (RIL) in the special economic zone (SEZ) at Jamnagar. The main objective behind setting up of RPL was to take advantage of the emerging opportunities in the energy sector i.e. the rising demand in the west for high quality fuels that meet stern emission rules and expanding price gap between heavy and light crude oil. In 2006, RPL made initial public offering (IPO) of 1350 million equity shares with a face value of Rs. 10. Out of it 450 million shares were offered for retail investors with a price band of Rs 57 to Rs 62. Retail investors had the option of paying Rs 16 per share on application. Shares were oversubscribed close to 50 times and finally shares were offered at Rs. 60, Rs 2 below the upper price band. RIL subscribed to an additional 900 million equity shares at Rs. 62. RIL was holding around 80 percent of stake in RPL in the post IPO period which also included the 2700 million shares subscribed at Rs. 10 before the public issue. Analysts saw positives about investing in RPL IPO mainly from the point of view of Reliance groups experience in running large capacity businesses. But they also cautioned about possible delay in commissioning of the refinery by the set deadline of December 2008. Concern was also raised about the fact that opportunity spotted by RPL was known to other MNCs as well and it may be just a matter of time before others get in to the same business. They also cautioned about the possibility of merger of new entity with RIL as they were in similar business and given the history of Reliance group of merging the subsidiaries with itself. RPL was set up in the special economic zone in Jamnagar, Gujarat, adjacent to the existing RIL refinery. The refinery had the capacity to process 580,000 barrels per day, or 27 million tonnes per annum of crude oil. The expected capital cost of the refinery was Rs. 270000 million. Company made considerable progress in 2007-08 in implementing the refinery project. Directors report on March 2008 claimed that it completed around 90 percent of the implementation of the project. It was one of the largest and most efficient refineries in the world with good synergies. In December 2008 it started the refining activity and announced that it will attain the full capacity shortly. With the completion RPL refinery, Jamnagar became the largest refining complex with an aggregate refining capacity of 1.24 million barrels of oil per day in any single location in the world. RECENT QUARTERLY PERFORMANCE RPL reported the income of Rs 76390 million from operations for the quarter ended in June 2009 and net profit was at Rs. 1050 million. Gross refining margin was $5.4 per barrel. 4.04 million tonnes of crude oil was processed in the quarter which included 20 types of crude oils. Total export was at 2.5 million tonnes of refined products to 26 countries. Company was successful in commissioning all key processing units in that quarter. RPL reported income of 36780 million for the year ended March 2009 and profit stood at 2270 million. Operating profit margin was 6.2 percent. MERGER On February 27, 2009 it was announced that RIL and RPL will be merged. Board meeting of RPL and RIL on March 2nd 2009 finalized the modalities of merging RPL with RIL. Swap ratio was fixed at one RIL share for 16 RPL shares (1:16). Decision of merger did not come as a surprise for market participants given the track record of Reliance group in the last three decades. Reliance group is a combination of several companies which were floated for executing specific projects and subsequently merged with it. (For e.g. Reliance Petrochemicals Ltd., Reliance Polyethylene Ltd. and Reliance Polypropylene Ltd.). Market observers argued that the strategy of Reliance group was to start large capital intensive projects in the balance sheet of a new company and once project is operational, merging them with the RIL. Benefit for RIL is that it is protected from risk of project execution as well as its balance sheet will not be affected by equity or debt burden. Different views have been expressed by analysts. It was argued that there was no business synergy in the merger of RPL with RIL. At least in the first merger of RPL and RIL in 2002, RPL was supplying refinery by- products to RIL. But in second merger in 2009, there was no difference between the refineries of RIL and RPL except for the technological superiority of RPLs refinery. The main reason for setting up of separate company (i.e. RPL) was to reap the benefits of special economic zone since such benefits were not available for expansion plans of the existing companies (Srinivasan, R. 2009). PTI reported that, it is an action replay after seven years. Announcement time as well as negative market reaction following merger announcement were common factors in both the mergers. Merger announcement timing was criticized on the ground that it did not provide any arbitrage opportunity for traders. Merger decision was first announced on Friday (27th February, 2009) after the market close and swap ratio (1RIL: 16RPL) was fixed before the market open next Monday (2nd March, 2009). Swap ratio was same as the ratio of the closing price of the stocks on the announcement day. Traders were not able to adopt strategies according to their analysis of the companies and speculation on swap ratio. On March 3rd, 2009, Economic Times reported that swap ratio of 1 RIL: 16 RPL was positive for RPL share holders and disappointing to RIL share holders. It was expressed by 3.1 percent fall in the RIL price and 1.4 percent fall in RPL price following the merger announcement. On same day The Hindu reported that merger was attractive for both RPL and RIL share holders. Merger helps RIL to consolidate its position since it acquired a world class refinery with minimum project risk; on the other hand, it helps RPL by reducing volatility in earnings. FUTURE PROSPECTS IN REFINING SECTOR: India has achieved impressive growth in refining sector since independence. It started with 0.25 million tonnes per annum at Digboi oil refinery, only refinery of independent India. As on December 2009, there are 20 refineries (17 in public sector and 3 in the private sector) with capacity of 179.956 MMTPA. India is not only self sufficient in refining oil for domestic consumption but also exports the petroleum products. Demand for petroleum products are directly linked to the energy requirements of the country and India being second fastest growing country in the world, shows positive signs for the refining sector. Report of the Working Group on Petroleum and Natural Gas for the XI Plan (2007 2012) states that refinery sector in Asia is turning attractive due to no substantial capacity addition in Europe. With only marginal capacity addition in US and in Central Asia refineries being old require large investment. Only Middle East, China and India have seen significant growth in refining sector. As Asia in general being projected as the growth centre for coming decades provides an excellent opportunity for Asian countries in general and India in particular. The feasible way would be establishing refineries which meet the world fuel standards and accessing world markets by exporting the surplus products. Report also stated that the target of 10th Five Years Plan (2002-2007) has been adequately met. 11th plan (2007- 2012) has fixed the target of 91.99 MMTPA capacity addition, however it states that actual capacity addition depends on various factors like domestic demand, tariff, refining margin and export potential. Plan expects the refining capacity to be around 190 to 200 MMTPA with the scope of export of 45 to 55 MMTPA by 2012. 11th plan has also given projection for the 12th plan period as well. An approximate assessment puts the capacity addition of 67.24 MMTPA (43.30 MMTPA in public sector and 23.94 MMTPA in private sector) is expected. But it depends on the commercial viability of the projects as well as demand for petroleum products during the 11th plan period. THE STOCK MARKET PERSPECTIVE According to stock market analysts, the share price of a company usually provided a true reflection of the companys present and expected financial performance. The stock price usually reflected various risks associated with the company, which could be broadly categorized as systematic and unsystematic risks (Refer Exhibit V). An analysis of the stock price performance of MRPL and RPL would help investors analyze the quantum of returns offered to them and identify the extent of risks associated with these companies over a specified period of time. The quarterly share prices of MRPL and RPL from January 2005 to May 2010 are provided in Table III to help measure the risks and returns of these two companies.

Queuing Theory Free Essays

string(51) " be exponentially distributed with mean 3 minutes\." Waiting Line Models ? ? ? ? ? ? ? ? The Structure of a Waiting Line System Queuing Systems Queuing System Input Characteristics Queuing System Operating Characteristics Analytical Formulas Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times Multiple-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times Economic Analysis of Waiting Lines Slide 1 Structure of a Waiting Line System ? ? Queuing theory is the study of waiting lines. Four characteristics of a queuing system are: †¢the manner in which customers arrive †¢the time required for service the priority determining the order of service †¢the number and configuration of servers in the system. Slide 2 Structure of a Waiting Line System ? ? Distribution of Arrivals †¢Generally, the arrival of customers into the system is a random event. We will write a custom essay sample on Queuing Theory or any similar topic only for you Order Now †¢Frequently the arrival pattern is modeled as a Poisson process. Distribution of Service Times †¢Service time is also usually a random variable. †¢A distribution commonly used to describe service time is the exponential distribution. Slide 3 Structure of a Waiting Line System ? Queue Discipline †¢Most common queue discipline is first come, first served (FCFS). An elevator is an example of last come, first served (LCFS) queue discipline. †¢Other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first. Slide 4 Structure of a Waiting Line System ? Single Service Channel Customer arrives ? Waiting line Multiple Service Channels System S1 Customer leaves System S1 Customer arrives Waiting line S2 Customer leaves S3 Slide 5 Examples of Internal Service Systems That Are Queueing Systems Type of System Customers Server(s) Secretarial services Employees Secretary Copying services Employees Copy machine Computer prog ramming services Employees Programmer Mainframe computer Employees Computer First-aid center Employees Nurse Faxing services Employees Fax machine Materials-handling system Loads Materials-handling unit Maintenance system Machines Repair crew Inspection station Items Inspector Production system Jobs Machine Semiautomatic machines Machines Operator Tool crib Machine Clerk Slide 6 Examples of Transportation Service Systems That Are Queueing Systems Type of System Customers Server(s) Highway tollbooth Cars Cashier Truck loading dock Trucks Loading crew Port unloading area Ships Unloading crew Airplanes waiting to take off Airplanes Runway Airplanes waiting to land Airplanes Runway Airline service People Airplane Taxicab service People Taxicab Elevator service People Elevator Fire department Fires Fire truck Parking lot Cars Parking space Ambulance service People Ambulance Slide 7 Queuing Systems ? ? ? ? A three part code of the form A/B/k is used to describe various queuing systems. A identifies the arrival distribution, B the service (departure) distribution and k the number of channels for the system. Symbols used for the arrival and service processes are: M – Markov distributions (Poisson/exponential), D – Deterministic (constant) and G – General istribution (with a known mean and variance). For example, M/M/k refers to a system in which arrivals occur according to a Poisson distribution, service times follow an exponential distribution and there are k servers working at identical service rates. Slide 8 Queuing System Input Characteristics = 1/? =  µ= 1/ µ = = the average arrival rate the average time between arrivals the average service rate for each server the average service time the standard deviation of the service time Slide 9 Queuing System Operating Characteristics P0 = Pn = Pw = Lq = probability the service facility is idle robability of n units in the system probability an arriving unit must wait for service average number of units in the queue awaiting service L = average number of units in the system Wq = average time a unit spends in the queue awaiting service W = average time a unit spends in the system Slide 10 Analytical Formulas ? ? For nearly all queuing systems, there is a relationship between the average time a unit spends in the system or queue and the average number of units in the system or queue. These relationships, known as Little’s flow equations are: L = ? W and Lq = ? Wq Slide 11 Analytical Formulas ? ? When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following: †¢M/M/1 †¢M/M/k †¢M/G/1 †¢M/G/k with blocked customers cleared †¢M/M/1 with a finite calling population Analytical formulas are not available for all possible queuing systems. In this event, insights may be gained through a simulation of the system. Slide 12 M/M/1 Queuing System ? ? ? ? ? ? Single channel Poisson arrival-rate distribution Exponential service-time distribution Unlimited maximum queue length Infinite calling population Examples: †¢Single-window theatre ticket sales booth Single-scanner airport security station Slide 13 Notation for Single-Server Queueing Models ? ? = Mean arrival rate for customers = Expected number of arrivals per unit time 1/? = expected interarrival time ? m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time 1/m = expected service time ? r = the utilization factor = the average fraction of time that a server is busy serving customers = /? m Slide 14 ? Assumptions 1. Interarrival times have an exponential distribution with a mean of 1/?. 2. Service times have an exponential distribution with a ean of 1/m. 3. The queueing system has one server. †¢ The expected number of customers in the system is L = r? /? (1 –? r) = /? (m? – ? )? †¢ The expected waiting time in the system is W = (1 / ? )L = 1 / (m – ? ) †¢ The expected waiting time in the queue is Wq = W – 1/m = ? / [m(m – ? )] †¢ The expected number of customers in the queue is Lq = ? Wq = ? 2 / [m(m – ? )] = r2 / (1 – r) Slide 15 ? The probability of having exactly n customers in the system is Pn = (1 – r)rn Thus, P0 = 1 – r P1 = (1 – r)r P2 = (1 – r)r2 : : ? The probability that the waiting time in the system exceeds t is P(W ; t) = e–m(1–r)t for t ? ? The probability that the waiting time in the queue exceeds t is P(Wq ; t) = re–m(1–r)t for t ? 0 Slide 16 Problem: ? Consider the situation where the mean arrival rate is one customer every 4 minutes and the mean service time is 2. 5 minutes. Calculate the following †¢Average no. of customer in the system †¢Average queue length †¢Average time a customer spends in the system †¢Average time a customer waits before being served. Slide 17 Problem: ? ? ? Arrivals at a telephone booth are considered to be Poisson, with an average time of 10 minutes between one arrival and the next. The length of a phone call is ssumed to be exponentially distributed with mean 3 minutes. You read "Queuing Theory" in category "Essay examples" What is the probability that a person arriving at the booth will have to wait? The telephone department will install a second booth when convinced that an arrival would expect to have to wait at least three minutes for the phone . By how much must the flow of arrivals be increased in order to justify a second booth? Slide 18 Example: SJJT, Inc. (A) ? M/M/1 Queuing System Joe Ferris is a stock trader on the floor of the New York Stock Exchange for the firm of Smith, Jones, Johnson, and Thomas, Inc. Stock transactions arrive at a mean rate of 20 per hour. Each order received by Joe requires an average of two minutes to process. Orders arrive at a mean rate of 20 per hour or one order every 3 minutes. Therefore, in a 15 minute interval the average number of orders arriving will be ? = 15/3 = 5. Slide 19 Example: SJJT, Inc. (A) ? Arrival Rate Distribution Question What is the probability that no orders are received within a 15-minute period? Answer P (x = 0) = (50e -5)/0! = e -5 = .0067 Slide 20 Example: SJJT, Inc. (A) ? Arrival Rate Distribution Question What is the probability that exactly 3 orders are received within a 15-minute period? Answer P (x = 3) = (53e -5)/3! 125(. 0067)/6 = . 1396 Slide 21 Example: SJJT, Inc. (A) ? Arrival Rate Distribution Question What is the probability that more than 6 orders arrive within a 15-minute period? Answer P (x ; 6) = 1 – P (x = 0) – P (x = 1) – P (x = 2) – P (x = 3) – P (x = 4) – P (x = 5) – P (x = 6) = 1 – . 762 = . 238 Slide 22 Example: SJJT, Inc. (A) ? Service Rate Distribution Question What is the mean service rate per hour? Answer Since Joe Ferris can process an order in an average time of 2 minutes (= 2/60 hr. ), then the mean service rate,  µ, is  µ = 1/(mean service time), or 60/2. m = 30/hr. Slide 23 Example: SJJT, Inc. (A) ? Service Time Distribution Question What percentage of the orders will take less than one minute to process? Answer Since the units are expressed in hours, P (T ; 1 minute) = P (T ; 1/60 hour). Using the exponential distribution, P (T ; t ) = 1 – e- µt. Hence, P (T ; 1/60) = 1 – e-30(1/60) = 1 – . 6065 = . 3935 = 39. 35% Slide 24 Example: SJJT, Inc. (A) ? Service Time Distribution Question What percentage of the orders will be processed in exactly 3 minutes? Answer Since the exponential distribution is a continuous distribution, the probability a service time exactly equals any specific value is 0 . Slide 25 Example: SJJT, Inc. (A) ? Service Time Distribution Question What percentage of the orders will require more than 3 minutes to process? Answer The percentage of orders requiring more than 3 minutes to process is: P (T ; 3/60) = e-30(3/60) = e -1. 5 = . 2231 = 22. 31% Slide 26 Example: SJJT, Inc. (A) ? Average Time in the System Question What is the average time an order must wait from the time Joe receives the order until it is finished being processed (i. e. its turnaround time)? Answer This is an M/M/1 queue with ? = 20 per hour and m = 30 per hour. The average time an order waits in the system is: W = 1/( µ – ? ) 1/(30 – 20) = 1/10 hour or 6 minutes Slide 27 Example: SJJT, Inc. (A) ? Average Length of Queue Question What is the average number of orders Joe has waiting to be processed? Answer Average number of orders waiting in the queue is: Lq = ? 2/[ µ( µ – ? )] = (20)2/[(30)(30-20)] = 400/300 = 4/3 Slide 28 Example: SJJT, Inc. (A) ? Utilization Factor Question What percentage of the time is Joe processing orders? Answer The percentage of time Joe is processing orders is equivalent to the utilization factor, ? /m. Thus, the percentage of time he is processing orders is: ?/m = 20/30 = 2/3 or 66. 67% Slide 29 Example: SJJT, Inc. A) Solution ? 1 2 3 4 5 6 7 8 9 A B C D E F Poisson Arrival Rate Exponential Service Rate Operating Characteristics Probability of no orders in system Average number of orders waiting Average number of orders in system Average time an order waits Average time an order is in system Probability an order must wait G ? m H 20 30 Po Lg L Wq W Pw 0. 333 1. 333 2. 000 0. 067 0. 100 0. 667 Slide 30 M/M/k Queuing System ? ? ? ? ? ? Multiple channels (with one central waiting line) Poisson arrival-rate distribution Exponential service-time distribution Unlimited maximum queue length Infinite calling population Examples: Four-teller transaction counter in bank †¢Two-clerk returns counter in retail store Slide 31 1 ? P? n ? m ? P0 , for (n ? k) ? n! ? ? n ? ? m ? P0 , for (n ? k) ? ? ? 1 n k ? 1 1 km ? ? ? ? n! ? m ? ? k! ? m ? km ? ? ? ? ? 1 ? k! k n ? k P? 0 P w ? n ? k ? 1 ? n ? 0 ? n 1 ? ? P(n ? k ) ? ?m? ? k! ? ? k km P0 , km ? ? k ?m ? ? m ? ? ? ? ? ? L? P0 ? 2 m (k ? 1)! (km ? ? ) W? L ? , Lq ? ,r ? km Lq ? 1 ? L? , Wq ? W ? ? m m ? Slide 32 General Operating Characteristics Little’ s F low Equations : L (or W ? ) ? Lq (or Wq ? ) ? L ? ?W L q ? ?Wq W ? Wq ? 1 m Slide 33 Problem: ? ? ? ? ? ? ? ? A Tax consulting firm has four service stations (counters) in its office to receive people who have problems and complaints about their income, wealth and sales taxes. Arrivals average 80 persons in an 8 hour service day. Each tax advisor spends irregular amount of time servicing the arrivals which have been found to have an exponential distribution. The average service time is 20 minutes. Calculate the average no. of customers in the system, average no. of customers waiting to be serviced, average time a customer spend in the system, average waiting time for a customer in queue. Calculate how many hours each week does a tax advisor spend erforming his job? What is the probability that a customer has to wait before he gets service? What is the expected no. of idle tax advisors at any specified time? Slide 34 Example: SJJT, Inc. (B) ? M/M/2 Queuing System Smith, Jones, Johnson, and Thomas, Inc. has begun a major advertising campaign which it believes will increase its business 50%. To handle the increased volume, the company has hired an additional floor trader, Fred Hanson, who works at the same speed as Joe Ferris. Note that the new arrival rate of orders, ? , is 50% higher than that of problem (A). Thus, ? = 1. 5(20) = 30 per hour. Slide 35 Example: SJJT, Inc. (B) ? Sufficient Service Rate Question Why will Joe Ferris alone not be able to handle the increase in orders? Answer Since Joe Ferris processes orders at a mean rate of  µ = 30 per hour, then ? =  µ = 30 and the utilization factor is 1. This implies the queue of orders will grow infinitely large. Hence, Joe alone cannot handle this increase in demand. Slide 36 Example: SJJT, Inc. (B) ? Probability of n Units in System Question What is the probability that neither Joe nor Fred will be working on an order at any point in time? Slide 37 Example: SJJT, Inc. (B) ? Probability of n Units in System (continued) Answer Given that ? = 30,  µ = 30, k = 2 and (? / µ) = 1, the probability that neither Joe nor Fred will be working is: 1 P0 ? k ? 1 ( ? / m )n (? / m ) k km ? ( ) ? n! k! km ? ? n? 0 = 1/[(1 + (1/1! )(30/30)1] + [(1/2! )(1)2][2(30)/(2(30)-30)] = 1/(1 + 1 + 1) = 1/3 = .333 Slide 38 Example: SJJT, Inc. (B) ? Average Time in System Question What is the average turnaround time for an order with both Joe and Fred working? Slide 39 Example: SJJT, Inc. (B) ? Average Time in System (continued) Answer The average turnaround time is the average waiting time in the system, W. Lq = ? µ(? / µ)k (k-1)! (k µ – ? )2 P0 = (30)(30)(30/30)2 (1! ((2)(30)-30))2 (1/3) = 1/3 L = Lq + (? / µ) = 1/3 + (30/30) = 4/3 W = L/ (4/3)/30 = 4/90 hr. = 2. 67 min. Slide 40 Example: SJJT, Inc. (B) ? Average Length of Queue Question What is the average number of orders waiting to be filled with both Joe and Fred working? Answer The average number of orders waiting to be filled is Lq. This was calcula ted earlier as 1/3 . Slide 41 Example: SJJT, Inc. (B) ? Formula Spreadsheet 1 2 3 4 5 6 7 8 9 10 A B C D E F Number of Channels Mean Arrival Rate (Poisson) Mean Service Rate (Exponential ) Operating Characteristics Probability of no orders in system Average number of orders waiting Average number of orders in system Average time (hrs) an order waits Average time (hrs) an order is in system Probability an order must wait G k ? m H 2 30 30 Po =Po(H1,H2,H3) Lg ## L =H6+H2/H3 Wq =H6/H2 W =H8+1/H3 Pw =H2/H3 Slide 42 Example: SJJT, Inc. (B) ? Spreadsheet Solution 1 2 3 4 5 6 7 8 9 10 A B C D E F Number of Channels Mean Arrival Rate (Poisson) Mean Service Rate (Exponential ) Operating Characteristics Probability of no orders in system Average number of orders waiting Average number of orders in system Average time (hrs) an order waits Average time (hrs) an order is in system Probability an order must wait G k ? m H 2 30 30 Po Lg L Wq W Pw 0. 333 0. 333 1. 333 0. 011 0. 044 1. 000 Slide 43 Example: SJJT, Inc. (C) ? Economic Analysis of Queuing Systems The advertising campaign of Smith, Jones, Johnson and Thomas, Inc. (see problems (A) and (B)) was so successful that business actually doubled. The mean rate of stock orders arriving at the exchange is now 40 per hour and the company must decide how many floor traders to employ. Each floor trader hired can process an order in an average time of 2 minutes. Slide 44 Example: SJJT, Inc. (C) ? Economic Analysis of Queuing Systems Based on a number of factors the brokerage firm as determined the average waiting cost per minute for an order to be $. 50. Floor traders hired will earn $20 per hour in wages and benefits. Using this information compare the total hourly cost of hiring 2 traders with that of hiring 3 traders. Slide 45 Example: SJJT, Inc. (C) ? Economic Analysis of Waiting Lines Total Hourly Cost = (Total salary cost per hour) + (Tota l hourly cost for orders in the system) = ($20 per trader per hour) x (Number of traders) + ($30 waiting cost per hour) x (Average number of orders in the system) = 20k + 30L. Thus, L must be determined for k = 2 traders and for k = 3 traders with ? = 40/hr. nd m = 30/hr. (since the average service time is 2 minutes (1/30 hr. ). Slide 46 Example: SJJT, Inc. (C) ? Cost of Two Servers P0 ? 1 k ? 1 (? ? n? 0 / m )n ( ? / m ) k km ? ( ) n! k! km ? ? P0 = 1 / [1+(1/1! )(40/30)]+[(1/2! )(40/30)2(60/(60-40))] = 1 / [1 + (4/3) + (8/3)] = 1/5 Slide 47 Example: SJJT, Inc. (C) ? Cost of Two Servers (continued) Thus, Lq = ? µ(? / µ)k (k-1)! (k µ -? )2 P0 = (40)(30)(40/30)2 1! (60-40)2 (1/5) = 16/15 L = Lq + (? / µ) = 16/15 + 4/3 = 12/5 Total Cost = (20)(2) + 30(12/5) = $112. 00 per hour Slide 48 Example: SJJT, Inc. (C) ? Cost of Three Servers P0 ? 1 k ? 1 (? ? n? 0 / m )n ( ? / m ) k km ( ) n! k! km ? ? P0 = 1/[[1+(1/1! )(40/30)+(1/2! )(40/30)2]+ [(1/3! )(40/30)3(90/(90-40))] ] = 1 / [1 + 4/3 + 8/9 + 32/45] = 15/59 Slide 49 Example: SJJT, Inc. (C) ? Cost of Three Servers (continued) (30)(40)(40/30)3 Hence, Lq = (15/59) = 128/885 = . 1446 (2! )(3(30)-40)2 Thus, L = 128/885 + 40/30 = 1308/885 (= 1. 4780) Total Cost = (20)(3) + 30(1308/885) = $104. 35 per hour Slide 50 Example: SJJT, Inc. (C) ? System Cost Comparison 2 Traders 3 Traders Wage Cost/Hr $40. 00 60. 00 Waiting Cost/Hr $82. 00 44. 35 Total Cost/Hr $112. 00 104. 35 Thus, the cost of having 3 traders is less than that of 2 traders. Slide 51 How to cite Queuing Theory, Essay examples