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Case Study: Manpower Resource Planning 65
distributions that were independently and identically distributed with mean service
rate μ i . Such a system is commonly known as the M/G/s queuing system (s > 1
finite). The mean total waiting time for the entire drug dispensing process in the
pharmacy can be computed by summing the mean waiting and mean service times
at each service station.
In order to derive the mean waiting times at each service station, the overall arrival
rate λ i at each service station, for the queuing network shown in Figure 5.2 has to be
computed. At statistical equilibrium, λ i is given by

N

λ i = λ i0 + λ j P ji ,
j=1
where N is the number of queuing stations in the network, λ i is the arrival rate at
station i from external sources, and p ji is the probability that a job is transferred to the
jth node after service is completed at the ith node.
In order to compute the mean waiting times at each single-server queuing stations
(s i = 1), the mean queue length of each single server-station, L ¯ M/G/1 , is first computed
i
with the well-known Pollaczek--Khintchine formula, 28
ρ 2 i 1 + CV i 2
M/G/1
L ¯ = .
i
1 − ρ i 2
2
Here CV is the squared cofficient of variation of service times, T i , which follow any
i
general random distribution:
Var(T i )
2
CV = ,
i ¯ 2
T
i
¯
In which Var(T i ) is the variance of the random service time of server i, and T i is the
mean of the service time of server i (or reciprocal of the service rate μ i ). ρ i is the
utilization of server i, which can be interpreted as the fracttion of time during which
the server is busy and is given by

λ i
ρ i = .
μ i
¯
Given L ¯ M/G/1 , the mean waiting times at each single-server service station, W M/G/1 ,
i i
28
can then be computed using Little’s theorem as follows:
L ¯ i M/G/1
M/G/1
¯
W = .
i
λ i
To compute the waiting times of queuing stations with multiple server (s i > 1),
we apply an approximation due Cosmetatos. 29 For this, we first compute the mean