Application of Queuing Theory to Waiting Time of Out-Patients in Hospitals.

Transcription

1 Applcaton of Queung Theory to Watng Tme of Out-Patents n Hosptals. R.A. Adeleke *, O.D. Ogunwale, and O.Y. Hald. Department of Mathematcal Scences, Unversty of Ado-Ekt, Ado-Ekt, Ekt State, Ngera. E-mal: waleod@yahoo.com ABSTRACT Ths paper consders the watng of patents n unversty health centers as a sngle-channel queung system wth Posson arrvals and exponental servce rate where arrvals are handled on a frst come frst serve bass. Hence, the m/m/ queung system s however proposed. The average number of patents, the average tme spent by each patent as well as the probablty of arrval of patents nto the queung system wll be obtaned. (Keywords: sngle channel queung system, Posson arrvals, exponental servce rates) INTRODUCTION The exstence of any naton s a functon of the survval of ts ctzens and n turn a functon of adequate health care programs of ts ctzenry. Health, no doubt has a great nfluence on the economy. As a result, the government establshed hosptals, prmary health care centers, federal medcal centers, unversty teachng hosptals, and so on, to mprove the health care of Ngerans at affordable cost. Also, several health polces and laws were formulated and adopted to make these aforementoned centers functonal. These health polces also brought about the establshment of global and local health organzaton such as World Health Organzaton (WHO). Unted Natons Internatonal Chldren Emergency Fund (UNICEF) and a host of others. Despte all these efforts, t should be noted that, there are stll some avodable problems whch undermne ther success of ths sector. One of the most frequent of them s the problem of watng lnes (queues) found n hosptals. Queung s a very volatle stuaton whch cause unnecessary delay and reduce the servce effectveness of establshments. Apart from the tme wasted, t also leads to breakdown of law and order. Many lves and property had been lost n queues at fllng statons n the past. Queues n hosptals often have severe consequences. For nstance, delay n treatment of asthma, dabetes, and cardac dsease patents often lead to complcatons and eventual death. (The World Bank Illustrated Home Medcal Encyclopeda, 998). In lght of ths, there s need for a crtcal evaluaton of patent watng tme as well as reducng or elmnatng t. METHODOLOGY Data on arrval tmes, tme servce begns, tme servce ends, and departure tme of 00 patents was collected over 4 days. Ths data wll enable us to obtan the arrval rate, the servce rate, and the traffc ntensty of the patents usng results from the brth and death model (whch s synonymous to arrval and departure). MODEL SPECIFICATION The m/m/ Queue (Sngle-Channel Queung System). In ths queung system, the customers arrve accordng to a Posson process wth rate. The tme t takes to serve every customer s an exponental random varable wth parameter μ. The servce tmes are mutually ndependent and further ndependent of the nter arrval tmes. When a customer enters an empty system, hs servce starts at once and f the system s nonempty, the ncomng customer jons the queue. The Pacfc Journal of Scence and Technology 70

2 When a servce completon occurs, a customer from the queue f any, enters the servce faclty at once to get served. THEOREM The process ( X( t), t 0) process wth brth rate death rate μ μ. s a brth and death 0 and Proof Because of the exponental dstrbuton of the nter arrval tmes and of the servce tmes, t s obvous that ( X( t), t 0) s a Markov process. On the other hand, snce the probablty of havng two events (departure and arrval) n the nterval f tme (, tt+ h) s 0( h). We therefore have: PXt [ ( + h) + Xt ( ) ] h+ 0( h) 0 PXt [ ( + h) Xt ( ) ] μh+ 0( h) PXt [ ( + h) Xt () ] ( + μ) h+ 0( h) PXt [ ( + h) Xt ( ) ] h+ 0( h), 0 PXt [ ( + h) Xt ( ) ] 0( h), j () Ths shows that ( X( t), t 0) s a brth and death process. Suppose π (), 0s the densty functon of the number of customers n the system at a steady-state, the balanced equaton for ths brth and death process s: π(0) μ π() Generally, () ( + μπ ) ( ) π( ) + μπ( + ) We defne μ (3) (4) referred to as the traffc ntensty called the mean quantty of work brought to the system per unt tme. A drect applcaton of (4) yelds a statonary queue-length densty functon of an m/m/ queue gven by: π() ( ), 0, < (5) Therefore, the stablty condton < smply says that the system s stable f the work that s brought to the system per unt tme s strctly smaller than the processng rate. It s noteworthy that the queue wll empty nfntely many tmes when the system s stable. That s, 0 from (5) π (0) > 0 (6) We observe that (5) whch s the densty functon of the queue length n a steady-state s a geometrc dstrbuton. We then fnd, Ex ( ) (7) called the mean number of customers. The Pacfc Journal of Scence and Technology 7

3 We also observe that Ex ( ) as, so that n practce f the system s not stable, then the queue wll explode. We also fnd from (5) that: V( x) ( ) (8) Agan, we fnd from (5) that the probablty that the queue exceed say customers n steady-state s Px ( ) (9) We adopt a sngle-channel queung system (m/m/) wth Posson arrvals and exponental servce rate and arrvals are handled on a frst come frst serve bass. In ths queung system, the average arrval rate s less than the average servce rate (.e., < μ ). If < μ, there would be an unendng queue. The followng formulas are developed for ths system. The average number of customers on the queue at any gven tme s: t (0) The average number of customers watng to be served at any tme t s: () The average number of customers n the system s: () The average tme n queue (before servce s rendered) s: μ( ) (3) The average tme n the system (on queue and recevng servce) s: μ( ) RESULTS (4) The arrval tmes as well as the tme servce began and ended for 00 patents n the Unversty of Ado-Ekt, Health Center were observed and recorded between 8.00am and 5.00pm. A total of 4days were used for the data collecton. The watng tmes and servce tmes were obtaned by subtractng arrval tme from the tme servce began for each day. Smlarly, servce tme was found by subtractng tme servce began from when t ended. Therefore: Total watng tmes 945 mnutes Total servce tmes 798 mnutes taken place n 945 mns and servce was rendered n 798 mns. Ths follows the arrval of 00 patents We arrve at the followng;. The arrval rate total of patents total watng tmes The Pacfc Journal of Scence and Technology 7

4 . The servce rate total no of patents 00 μ 0.53 total servce tmes Thus, traffc ntensty μ 4. The average number of patents queue The average number of patents n queue when queue exst The average number of patents n the systems The average tme n queue [before servce s rendered mn utes μ( ) Alternatvely, we calculate t as Average no of patents n the system the servce rate mn utes The average tme of the system (on queue and recevng treatment) mn utes μ( ) 9. Average tme of servce Average tme n the system average tme n queue 8 mnutes 0. The probablty of queueng on arrval Traffc ntensty μ. The probablty of not queung on arrval n. The probablty that there are n patent n the system ( ) patents n the systems Hence, the probablty that there are The Pacfc Journal of Scence and Technology 73

5 n 3. The probablty that there are more than n patents n the system s. Hence, the probablty that there are more than patents n the system s DISCUSSION OF RESULTS The traffc ntensty s the probablty of patents queung on arrval whch clearly ndcates a hgher possblty of patents watng for treatment snce the doctor s busy renderng servce to a patent that has earler arrved. Ths also shows that the servce n the health centre s not 00% effcent. It then follows that there wll always be queue snce result (8) s greater than (7). That s the average tme n the queue system (both on queue recevng servce) s greater than the average tme n queue before servce s rendered. CONCLUSION AND RECOMMENDATION The queue theory s a useful statstcal technque for solvng pecular problems. Its applcatons n the organzaton are ndspensable. The queung problems encountered at the Unversty of Ado- Ekt Health Centre s smlar to what s encountered n older centers as well as other government hosptals across the country. Excessve waste of tme n the hosptals or health centers may lead to patents health complcatons and n some cases eventual death whch may be avoded. n attendng to patents and hence the servce effcency. REFERENCES. Churchman, G.W., R.C. Ackoff, and F.C. Arnoff Introducton to Operaton Research. John Wlley and Sons: New York, NY. pp Ignzo, J.P. 98. Lnear Programmng n Sngle and Multple objectve Systems. Prentce-Hall Eagle Wood Clffs, NJ. 3. Adeleke, R.A., C.E. Adeby, and O. Aknyem. 006l Applcaton of Queueng theory to Omega Bank PLC, Ado- Ekt, Ngera. : Taha, H.A Operaton Research: An Introducton, Fourth Edton. Macmllan Publshng: New York, NY. pp SUGGESTED CITATION Adeleke, R.A., O.D. Ogunwale, and O.Y. Hald Applcaton of Queung Theory to Watng Tme of Out-Patents n Hosptals. Pacfc Journal of Scence and Technology. 0(): Pacfc Journal of Scence and Technology As a result, t s recommended that more doctors should be deployed to these centers so as to convert the sngle-channel queung unts to multchannel queung unts. Ths wll offer servce on arrval. It s also recommended that more health care centers should be created to take care of all categores of patents (students or members of staff) n the unversty communty. Agan, more paramedcal offcers should be deployed to these centers. Ths wll take care of patents prelmnary tests or servce before they see the doctors. Ths wll reduce the servce tme spent by the doctors The Pacfc Journal of Scence and Technology 74