The document discusses a queuing model related to service rates and customer arrivals, particularly in a single-server context (M/M/1). It analyzes mean arrival rates, expected queue lengths, and waiting times based on given service patterns, specifically focusing on a mechanic's workload and idle time during an eight-hour workday. Additionally, it calculates the number of jobs expected at the start of each day under these conditions.

1. @Stoct socieing
Mode! I :(M/M/1)(oo /FCFs) os fointh amd Death Madel
A Mean arsival Tate (mo. o arival er unit time)
= Meon Sesvice ate (no.o customer sesved pes unit time)
oTsokic tmtenstH
he system Ls
Ebected (avesage) no. o unets m 4he Sstem Ls
Ls .
ud Epec ted (average) uue lemgth Le
L=Ls
(H-)
ivExheched tloiting
the queua, Na,
Um

2. Exhected daing Line n he System, Ws
@Stoet PaocBicing
w Expected
(u Expected waikingtime oh a Customea who has to wait (w/w7o)
(w/W>o)= -
(v) Expected lengBh omom-emt 9,uua
(LJL7O) =
u-a
m) Fsobabi lH4 ob Waiting time in the que 2t
on) Paobabi lidg
a ( 1 - a ) e ( K - 3 ) n ,
dW
u a u e Size 2 N = f"
P(
a)
e"(4-
a)w
dw

3. @stavd aacticung
(Poobabilik (laiking tine n he system 2 )
f4-2)e
(H-A)w
dw
t
Ltles forvmulas
Ls A Ws
a,=s

4. Stoct Psocicing
Q: A
exponental distibution with
:A TV. me chamic fnds that ha tume spent on his Tobs has an
Tv.
mean 30 muinutes, ih he epairs
Sets in the o7der in which hey Come in.
1 he
oisson with on
asoival o
aNexage gate o lo
mec haniS expected idle
day? Houw many Jobs ase ahead o he awexoge
sets is appsoximately
pea
me each
eight-hous da, what is the
da?
Just bouaht in ?
man
Sets
30
10
Solution : 48 48
30 8 X60
Expected no. ot Jobs
48
Ls -A L
48
30 5
L
3
Ls 48-30
s1Jobs. 30ky8

5. Stort Psacicun
the action o he teme the mechanicd is busy
the
Smce
emaums busy
he no.
ok hous o which the epaisman
erg bt- houwt day
in
8x =
5 houns
48
idle
tma toft ohich e mechamic
semain
me
m
PSemam
Therceph, the = 8 -5)hoors.
egh- hou day
= 3 hou»s.