Impatient customers-a factor of impact on
counter service organization
Marija Marinović
Faculty of Philosophy, Omladinska 14
HR-51000 Rijeka, e-mail:
Zdenka Zenzerović
Faculty of Maritime Studies at Rijeka, Studentska 2
HR-51000 Rijeka, e-mail:
Abstract. The subject of research in this paper was the counter service queuing system involving impatient customers since in real life queues with relatively great number of impatient customers can be met quite often, where some of them give up and leave the queue. The objective of this paper is to find out as to how impatient customers impact on the counter service organization. In order to realize the objective set, it was necessary to compare indicators of the queuing system with and without impatient customers. Special attention was paid to the loss probability, occurring in counter service queuing systems involving impatient customers, which shows the probability of leaving a queue by customers. Analysis of the parameters influencing the amount of the loss probability showed that its value can be reduced by changing parameters and thus impact on the performance of the counter service queuing system with impatient customers. The results of analysis of the queuing system with impatient customers are illustrated through the organization of a real life counter service.
Key words: queuing theory, queuing system with impatient customers, loss probability, counter service
1. Introduction
The paper [5] describes counter service system as a queuing system with an unlimited number of customers in a queue. The starting assumption was that the arrival rate was constant and that it did not depend on system state, i.e., on number of customers already in the system, either with servers or in queues.
However, in real life, customers who are faced with a number of other customers already waiting larger than a certain number, leave the system, i. e. give up the service. Customers who cannot or do not want to wait to be served in queue or servers are called impatient customers. So, there are impatient customers who leave either queue or servers.
The objective of this paper is to establish as to how impatient customers impact on counter service queuing system. In order to realize the task, counter service is defined as a queuing system involving impatient customers in queue. Queuing system indicators with and without impatient customers are then compared. Since the loss probability shows the quantity of impatient customers that probably will not access the counter, meaning that these are referred to as a "loss" of customers in each queuing system, it is important to explore as to how changes in certain parameters influence its value.
At the end a real life example of counter service queuing system and the results used for the assessment of counter service performance are analyzed.
2. Performance indicators of the queuing system with impatient customers
Considering the type of leaving from the system by customers, the paper considers the first case when customers in queuing system with impatient customers express impatience only when are in queue. When they approach the server, they patiently wait to be served, meaning that leaving from system after getting into service is not considered.
Observed queuing systems is based on the following assumptions:
- Queueing system has S service places and n customers in the system
- Arrival rate is a simple one with intensity λ
- Service rate is a simple one with intensity μ
- Average time for which a customer leaves the queue is
- Intensity of giving up waiting is υ.
Taking into consideration the fact that queue leaving influences the probability pn for nS, the following state probabilities are obtained from the Kolmogorov's equation [8]:
, , ..., (1)
(2)
(3)
,(4)
where:
- traffic intensity
- leaving coefficient
m-queue capacity (max. number of customers)
On the basis of formulas (1) and (3) and implementing equation stating that the sum of probabilities in all system states equals to 1, it follows:
(5)
From expression (5), the probability that there is no customers in the system is:
(6)
i.e.,
. (7)
This is the formula for probability p0 for queuing system with impatient customers.
In a queuing system without impatient customers of type M/M/S/ with infinite number of potential customers in a queue (m) probability p01 equals to [8]:
,(8)
i.e.:
(9)
where
,
Formulas (8) and (9) make sense for , i.e., when the average number of customers served in a unit of time at all service places is greater than average number of customers arriving to the queuing system in a unit of time.
The probability p0 shows the probability of service places being idle and it is an important indicator of any queuing system performance. Considering the subject of the research, it is necessary to compare probabilities p0 and p01 for queuing system both with and without impatient customers.
Formula (6) can be transformed in a shorter form as follows [8]:
(10)
where for m.
Comparison of probabilities p01 and p0 gives
p01 – p0= d
where p01 refers to systems without impatient customers, while p0 refers to system with impatient customers. Consequently
. (11)
From (11) follows:
.
Since
,
the following is valid
.
Since the numerator equals to > 0 i , and denominator to > 0 for <1, it follows i and d <0, it further follows that p0, i.e., the probability that queuing system with impatient customers is idle is greater than in queuing system without impatient customers.
This proves that probabilities p01 and p0 are not equal (for constant parameters S, and ) and that they depend on presence of impatient customers.
It is necessary to calculate values for particular indicators in order to perform the analyses of the performance of queuing system with impatient customers. This will obtained by using formulas [8,???]:
1) Expected number of not served customers
2) The probability of giving up service is proportional to the intensity of giving up waiting , and inversely proportional to intensity of arrival rate , hence:
3) Average number of customers waiting for service
4) Average number of customers being served
5) Average number of customers in the system
6) Absolute capacity of the system for service
,
7) Relative capacity for service
8) Average time spent in queue
9) Average time spent in system
.
Indicators for queuing systems with impatient customers, when implementing formula (6), can be compared with queuing systems without impatient customers in an analogous way. Results show that indicators: average number of customers in a queue, average number of customers in a system, average time spent in a queue and in a system are lower than in queuing system without impatient customers.
In queuing system with impatient customers, some users, after having waited for time τ, leave the system and become lost customers, which adversely impact any queuing system, so the counter service as well. Therefore, it is necessary to measure the possibility of occurring lost customers.
3.Loss probability
The performance measure Ploss is defined by: Ploss = the long-run fraction of customers who are lost [3]. The formula for the loss probabilitywas first implemented by Barrer in 1957, while Gnedenko and Kovalenko modified it in 1989. In their work [3], Bootss and Tijms, 1999. presented an alternative formula for the loss probability queuing system including multiple service places, as follows:
(12)
where
, (13)
and
. (14)
It follows from (12) that the loss probability depends on three parameters: number of serving places S, system utilization coefficient α and average time for which a customer leaves the queue .
In order to establish the impact of the loss probability on the performance and how to obtain optimal performance of a queuing system with impatient customers, impact of particular parameter to loss probability has been performed.
For that purpose, Tables 1, 2 and 3 have been drawn up, containing values for the loss probability of the chosen parameter values: number of service places (S), value of utilization system coefficient () and average time for which a customer leaves a queue ().
Table 1.Loss probability depending on number of service places (S) and average time for which a customer leaves a queue ()
Table 2.Loss probability depending on system utilization coefficient (α) and average time for which a customer leaves a queue ()
Table 3.Loss probability depending on number of service places (S) and utilization system coefficient (α)
Radi preglednosti the data from Table 1.-3. are presented in Figure 1.-3.
Figure 1.
Figure 2.
Figure 3.
From Table 1.-3. and Figure 1.-3. it can be concluded that loss probability decreases with:
- increase of number of service places (S)
- increase of time for which a customer leaves a queue ()
- decrease of utilization system coefficient (α).
The increase of service place number must not be arbitrary, rather, it must enable efficient performance of a system. Organization cannot considerably influence the increase of average time for which a customer leaves a queue, except for taking some marketing measures, rather, it depends on time available to customer i.e., his or her patience. Decrease of coefficient α can be obtained in two ways: increase of which is not cost efficient or increase of service rate.
4. Organization of counter service
U radu /../ promatran je proces opsluživanja jedne policijske postaje kao sustava masovnog opsluživanja s čekanjem i beskonačnim brojem stranaka u redu čekanja.
Međutim, budući da je pojava relativno dugačkih redova, općenito pa tako i u policijskoj postaji česta pojava, događa se da korisnici, ukoliko zateknu u redu za čekanje više korisnika od određenoga broja, napuštaju sustav, tj. odustaju od opsluživanja. U ovom radu, koji je nastavak istraživanja organizacije šalterske službe iz rada / /, proces opsluživanja na šalterima bit će razmatran kao sustav opsluživanja s nestrpljivim korisnicima, a teorijske postavke iz odjeljka 2. i 3. ovog rada bit će primijenjene na konkretan primjer šalterske službe jedne policijske postaje.
4.1. Problem description
The analysis of the impatient customer’s impact to queuing system efficiency was performed for an existing organization of Police Department counter service the offering following services:
Counter 1 - Issuing driving licenses and license plate
Counter 2 - Issuing ID and passports
Counter 3 - Registration/deregistration of residence.
Statistical observation in period from January 2, 2002 to May 2, 2002. produced data on total number of customers requiring one or more of above services or arrived at a counter, as follows:
Counter 1 - 6588 customers
Counter 2 - 1356 customers
Counter 3 - 1019 customers.
Considering working hours of six hour a day, the average number of customers per day and hour respectively is as follows:
Counter 1 - 77.51 customers/day; 12.91765 customers/hour
Counter 2 - 15.95 customers/day; 2.65882 customers/hour
Counter 3 - 11.99 customers/day; 1.99804 customers/hour.
It was also established that average processing time for any counter was 7.5 minutes. Budući da je riječ o opsluživanju s nestrpljivim korisnicima, statističkim promatranjem je utvrđeno da prosječno vrijeme napuštanja reda čekanja iznosi 6 minuta.
The objective is to make business decision, koja će, na temelju svih prethodno navedenih činjenica, that would enable optimal counter operation, i.e., working schedule providing customers with services in the shortest possible time and reducing idle time of clarks to minimum.
Usporedbom pokazatelja funkcioniranja obaju sustava, bez i sa nestrpljivim korisnicima, dobit će se ocjena uspješnosti njihovog funkcioniranja te optimalno rješenje organizacije šalterske službe.
4.2. Analysis and discussion of the solution
Since the counter service is defined as a queuing system, the problem has been solved using the queuing theory.
On the basis of adequate formulas, counter operation indicators have been calculated for one system without and for system with impatient customers, and shown in Table 4.
Table 4. Comparison of Police Department counter service operation indicators
without (A) and with (B) impatient customers
No. / Indicator / Unit / A / B1. / Arrival rate () / cust./hour / 17.57451 / 17.57451
2. / Service rate () / cust./hour / 8 / 8
3. / Traffic intensity () / - / 2.19681 / 2.19681
4. / Number of service
places (S) / counter / 3 / 3
5. / Utilization system coefficient (/S =α) / - / 0.73227 / 0.73227
6. / Average time for which a customer leaves a queue (τ) / hour / - / 0.1
7. / Intenzitet napuštanja nestrpljivih korisnika (υ) / cust./hour / - / 10
8. / Average number of customers in thequeue (LQ) / customer / 1.47845 / 0.247347
9. / Average number of customers in the system (L) / customer / 3.67527 / 2.13497
10. / Average time a customer spends in the queue (WQ) / hour / 0.08413 h
(5.05min) / 0.844min
11. / Average time a customer spends in the system (W) / hour / 0.20913 h (12.55min) / 0.121481 h
(7.2 min)
12. / Probability that all service places are idle (p0) / % / 8.19% / 11.34%
13. / Probability of loss custo-
mers (Ploss) / % / - / 9.61%
Prema rezultatima iz Tablice 4. proizlazi da je očekivani broj stranaka u redu čekanja s nestrpljivim korisnicima manji nego kod sustava bez nestrpljivih korisnika. Isti se zaključak odnosi i na ostale pokazatelje, tj. prosječan broj stranaka u sustavu, kao i prosječno vrijeme čekanja provedeno u redu i u sustavu. Međutim, vjerojatnost da će šalteri biti slobodni, tj. da stranka neće morati čekati je veća kod sustava s nestrpljivim korisnicima. Također se kod sustava s nestrpljivim korisnicima pojavljuje vjerojatnost izgubljenih korisnika Ploss.
Na temelju prethodnih rezultata zaključuje se da na uspješnost poslovanja šalterske službe najviše utječu pokazatelji p0 i Ploss. Vjerojatnost p0 predstavlja vjerojatnost neiskorištenog kapaciteta šaltera, a Ploss vjerojatnost “izgubljenih” korisnika. Prvi pokazatelj djeluje na povećanje troškova poslovanja, a drugi na smanjenje prihoda, što na kraju rezultira smanjenjem uspješnosti poslovanja šalterske službe.
Optimalno rješenje problema je broj šaltera koji se određuje na temelju vrijednosti tih vjerojatnosti koje su u praksi uobičajene i prihvatljive.
5.CONCLUSION
Proces opsluživanja na šalterima može se definirati kao queueing system te analizirati njegovo funkcioniranje using the queueing theory.
Budući da je u praksi česta pojava da zbog relativno dugačkih redova korisnici napuštaju red čekanja ukoliko je broj korisnika u redu veći od nekog uobičajenog, odnosno prihvatljivog broja, potrebno je uzeti u obzir tzv. nestrpljive korisnike.
Usporedbom pokazatelja funkcioniranja šalterske službe sa i bez nestrpljivih korisnika izlazi da u sustavu s nestrpljivim korisnicima pojedini pokazatelji (broj korisnika i duljina vremena čekanja) imaju manju vrijednost, što je povoljnije za korisnike. Međutim, povećava se vjerojatnost da će šalter biti slobodan, odnosno neiskorišten i pojavljuje vjerojatnost “izgubljenih” korisnika. Oba pokazatelja utječu na uspješnost poslovanja sustava opsluživanja, o čemu bi trebalo voditi računa pri organizaciji rada šalterske službe.
Optimalno rješenje problema bit će onaj broj šaltera za koji će vjerojatnosti nezauzetosti šaltera i izgubljenih korisnika poprimiti što manje vrijednosti.
REFERENCES
[1] N. K. BOOTS, H. TIJMS, A Multiserver Queueing System with Impatient Customers, Managament Science/Vol.45, No. 3, March 1999. 444-448.
[2] Z. ZENZEROVIĆ; M: MARINOVIĆ, Impact of service place specialization on the efficiency of queuing system functioning, Operational Research Proceedings KOI 2002, 311-320.
Analiza odabranog problema je proširena uzevši u obzir da šalterska služba može biti organizirana s višenamjenskimšalterima, kao što je pokazano u tablici 4. ili, pak, da su šalteri specijalizirani s obzirom na vrstu usluga koje pruža policijska postaja.
Vezano za rezultate rada [ ] uzete su u razmatranje dvije varijante organizacije šalterske službe:
1.multipurpose (universal) counters, meaning that any customer can be served at any counter,
- specialized counters, i.e., customers can be served at particular counter depending on service type required.
On the basis of adequate formulas, counter operation indicators have been calculated for one system of M/M/3/type (varijanta A), for one system M/M/2/type (varijanta B) and two systems M/M/1/type(varijante C i D), without and with impatient customers, and shown in Table 5., where is:
Variant A: multipurpose service places
Variant B: specialized service place for issuing driving licenses and license plate
Variant C: specialized service place for issuing ID and passports
Variant D: specialized service place for registration/deregistration of residence.
Vrijednosti u zagradama tablice 5. odnose se na nestrpljive korisnike.
Table 5.Comparison of Police Department counter service operation indicators
by counter type
No. / Indicator / Unit / A / B / C / D1. / Arrival rate () / cust./h / 17.57451 / 12.91765 / 2.65882 / 1.99804
2. / Service rate () / cust./h / 8 / 8 / 8 / 8
3. / Traffic intensity () / - / 2.19681 / 1.61471 / 0.33235 / 0.24975
4. / Number of service
places (S) / counter / 3 / 2 / 1 / 1
5. / Utilization system-coefficient (/S =α) / - / 0.73227 / 0.80735 / 0.33235 / 0.24975
6. / Average time for which a customer leaves the queue (τ) / hour / 0.1 / 0.1 / 0.1 / 0.1
7. / Intenzitet napuštanja nestrpljivih korisnika (υ) / cust./h / 10 / 10 / 10 / 10
8. / Average number of customers in thequeue (LQ) / customer / 1.47845
(0.247347) / 3.02284
(0.284062) / 0.16544
(0.042894) / 0.08314
(0.025016)
9. / Average number of customers in the system (L) / customer / 3.67527
(2.13497) / 4.63755
(1.543691) / 0.49780
(0.321629) / 0.33290
(0.243501)
10. / Average time a customer spends in the queue (WQ) / hour / 0.08413 h
5.05min
(0.844min) / 0.23400 h 14.04min
(1.32min) / 0.06223 h 3.73min
(1.07min) / 0.04161 h
2.5 min
(0.75min)
11. / Average time a customer spends in the system (W) / hour / 0.20913 h 12.55min
(7.2 min) / 0.35901 h
21.54min
(0.119502)
(7.17 min)] / 0.18723 h 11.23min
[0.120967] / 0.16612 h 9.97min
[0.013669
(0.82 min)]
12. / Probability that all service places are idle (P0) / % / 8.19%
[11.34%] / 10.66%
[20.48%] / 66.76%
[72.13%] / 75.02%
[78.15%]
13. / Probability of loss
customers (Ploss) / % / [9.61%] / [17.84%] / [13.90%] / [10.65%]
U sustavu s nestrpljivim korisnicima treba uzeti u obzir “izgubljene” korisnike, odnosno izračunati vjerojatnost Ploss. Za promatrane varijante proizlazi da je najveća vjerojatnost izgubljenih korisnika 17.84% za varijantu B sustava, za koji je koeficijent iskorištenja sustava 0.33235, intenzitet napuštanja sustava 10 korisnika/sat, a najmanja za varijantu A 9.61% s koeficijentom iskorištenja sustava 0.73227 i intenzitetom napuštanja sustava 10 korisnika/sat.
Thus, under given condition, the implementation of specialized counter in Police Department counter service is justified only if the service rate is at least 9.75024 customers/hour; therefore, service time should be reduced to 6.15 minutes, i.e., service rate should be increased by 21.9%.
Considering the service type offered at particular counter and proportion of human work required in providing a service, it is not always possible to reduce such service time.
However, assuming that in this example is possible to increase service rate to 9.75024 customers/hour, the indicator of expected number of customer in a queue is favorable, but increasing number of counters would severely impact particular counter idle time from 10.66% to 75.02% of total working hours, which is unacceptable in terms of economic considerations.
Due to the above, the combinations of multipurpose (universal) and specialized counter service organization have been considered as follows:
- Specialized counters for driving licenses (S1 and S2) and multipurpose counters for IDs and registration/deregistration of residence (S3 and S4)
- Specialized counters for driving licenses (S1 and S2) and multipurpose counter for IDs and registration/deregistration of residence (S3)
- Specialized counter for driving licenses (S1) and multipurpose counter for IDs and registration/deregistration of residence (S3).
Counter operation indicators as per above variants are shown in Table 3.
Table 3. Police Department counters operation indicators for counter combinations
No. / Indicator / Unit / Multipurposecounter S3 / Multipurpose counters
S3 i S4
1. / / cust./hour / 4.65686 / 4.65686
2. / / cust./hour / 8 / 8
3. / / - / 0.58211 / 0.58211
4. / S / counter / 1 / 2
5. / /S / - / 0.58211 / 0.29105
6. / LQ / counter / 0.81085 / 0.05388
7. / L / counter / 1.39296 / 0.63598
8. / WQ / hour / 0.29912
(17.95 min) / 0.13657
(8.19 min)
9. / W / hour / 0.17412
(10.45 min) / 0.01157
(0.69 min)
10. / P0 / % / 41.79 / 54.91
Variants 1 and 2 (due to rather large number of customers) that there will be two counters for driving license (S1 and S2), while other services will be handled either at counters S3 or S4, or only at counter S3, thus reducing the number of counters to three.
Variant 3 resulted from experience to date, where driving license were only handled at one counter. It is practicable if, on the basis of past number of customers as noted during observed period, average processing time ranges from 4 to 4.6 minutes, e.g. =15 customers/hour.
Table 4. shows results in comparison of indicators for proposed counter service variants.
Table 4. Comparison of indicators for combined counter service variants
Indicator / Variant 1. / Variant 2. / Variant 3.( = 15)
S
/ 4 / 3 / 2LQ / 2.12 / 2.88 / 6.15
P0 / 37.6% / 27.47% / 27.8%
On the basis of results shown in Table 4. and Table 2., one of the following business decisions should be made:
- Organize counter service as one system containing three multipurpose counters.
- Organize counter service with two counters, one specialized for driving licenses and the other multipurpose counter for other services.
1. Loss probabilityin respect to a constant value of utilization system coefficient (α) is decreased when the number of service places () and the average time for which a customer leaves a queue () are increased.